Slide 1 / 175 Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 175 Riemann Sums Trapezoid Approximation Area Under a Curve (The Definite Integral) Antiderivatives & The Fundamental Theorem of Calculus, part II Fundamental Theorem of Calculus, part I Average Value & Mean Value Theorem for Integrals Indefinite Integrals U-Substitution
Integration Slide 4 / 175 Mathematicians spent a lot of time working with the topic of derivatives, describing how functions change at any given instant. They then sought a way to describe how those changes accumulate over time, leading them to discover the calculation for area under a curve. This is known as integration, the second main branch of calculus. Finally, Liebniz and Newton discovered the connection between differentiation and integration, known as the Fundamental Theorem of Calculus, an incredible contribution to the understanding of mathematics. Slide 5 / 175 Riemann Sums Return to Table of Contents Area of Curved Shapes Slide 6 / 175 Formulas for the area of polygons, such as squares, rectangles, triangles and trapezoids were well known in many early civilizations. However, the problem of finding area for regions with curved boundaries (circles, for example) caused difficulties for early mathematicians. The Greek mathematician Archimedes proposed to calculate the area of a circle by finding the area of a polygon inscribed in the circle with the number of sides increased indefinitely.
Distance Using Graphs Slide 7 / 175 Consider the following velocity graph: v(t) (mph) 30 mph t (hours) 5 hrs How far did the person drive? Distance Using Graphs Slide 7 () / 175 Consider the following velocity graph: v(t) (mph) You can see that this number can 30 mph be obtained if we calculate the area under the velocity graph. So, the area of the rectangle in this case represents the total distance t (hours) traveled. 5 hrs How far did the person drive? Non-Constant Speed Slide 8 / 175 However, objects seldom travel at a constant speed. v(t) (mph) 50 mph 30 mph 5 hrs t (hours) The area under this graph is still equal to the distance traveled but we need more than just simple multiplication to find it.
Georg Friedrich Riemann Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to function analysis and found an approach for approximating the total area underneath a curve by dividing the total area into a series of rectangles. Slide 9 / 175 So, the area under the curve would be the sum of areas of the rectangles. Later, we will discuss how close this approximation is, and if there is any possibility to calculate the exact area underneath the curved boundary. Area Under the Curve Slide 10 / 175 Note: When we use the language "area under the curve" we are referring to the area between the function and the x-axis. Teacher Notes Let students discuss what the area would be if it wasn't bounded by the x- axis. Hopefully they will conclude that the area would be infinite. Area Under the Curve Note: When we use the language "area under the curve" we are referring to the area between the function and the x-axis. vs. Slide 10 () / 175 Finite Area Area
RAM - Rectangular Approximation Method Rectangular Approximation Method is a way to estimate area by drawing rectangles from the x-axis up to the curve. Slide 11 / 175 The question is: What part of the "top" of the rectangle should lie on the curve? Also, how many rectangles should be used? The right hand corner (RRAM) The left hand corner (LRAM) The middle (MRAM) Riemann Sums Slide 12 / 175 Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. Slide 13 / 175
Riemann Sums Slide 14 / 175 Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. Finally, calculate the area using LRAM. Have students discuss why this is an under approximation. Riemann Sums Slide 15 / 175 Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. We calculated the area using LRAM to be If we look at our graph, we can see that all of the rectangles are below our curve. Therefore, this approximation is an underestimate. Riemann Sums Slide 16 / 175 Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and right endpoints (RRAM). Is this approximation an overestimate or an underestimate?
Slide 16 () / 175 Riemann Sums Example: Approximate the area under the curve y = x 2 on [0,1] with a Riemann sum using 4 sub-intervals (rectangles) and midpoints (MRAM). Slide 17 / 175 Slide 17 () / 175
Riemann Sums Slide 18 / 175 *NOTE: MRAM LRAM + RRAM 2 Riemann Sums Slide 18 () / 175 Teacher Notes Technically, for some functions, MRAM could be the average of LRAM and *NOTE: RRAM; however, for most functions given in practice and on MRAM the exam, LRAM this will + RRAM not be the case. Have a discussion 2 with students about why this can't always be true. 1 When approximating the area under the curve y=3x+2 on [1,4] using four rectangles, how wide should each sub-interval be? Slide 19 / 175
Slide 19 () / 175 2 Approximate the area under y=3x+2 on [1,4] using six rectangles and LRAM. Slide 20 / 175 Slide 20 () / 175
3 Find the area under the curve on [-3,2] using five sub-intervals and RRAM. Slide 21 / 175 Slide 21 () / 175 Slide 22 / 175
Slide 22 () / 175 Slide 23 / 175 Slide 23 () / 175
Riemann Sum Notation The following notation is used when discussing Riemann sums and approximating areas. Some questions will use this notation, so it is important to be familiar with the meaning of each symbol. Slide 24 / 175 Using the symbols above, can you create a mathematical relationship between all 4 of them? Riemann Sum Notation The following notation end is of interval used when discussing Riemann sums and approximating areas. number Some of sub-intervals questions will use this notation, so it is important to be familiar width with of each the interval meaning of each symbol. Teacher Notes start of interval Some students may need guidance coming up with a relationship. The most common answer students will come up with is usually because they have been calculating the width in earlier questions. Slide 24 () / 175 Using the symbols above, can you create a mathematical relationship between all 4 of them? Slide 25 / 175
Slide 25 () / 175 Slide 26 / 175 Slide 26 () / 175
Riemann Sums with Tables Slide 27 / 175 Sometimes, instead of being given an equation for f(x), data points from the curve will be presented in a table. Provided the necessary information is in the table, you are still able to approximate area. Riemann Sums with Tables Example: Approximate the area under the curve, f(x), on [-2,4] using right endpoints and n=3. Slide 28 / 175 Slide 29 / 175
Riemann Sums with Tables Slide 30 / 175 Note: When using tabular data for Riemann Sums, not all subintervals need to be of equal width. If the question does not specify, then you are able to use the data provided - just make sure to account for the varying width. Riemann Sums with Tables Slide 31 / 175 Example: Using the subintervals in the table, approximate the area under using a left hand approximation. Riemann Sums with Tables Slide 32 / 175 Example: Using the subintervals in the table, approximate the area under using a right hand approximation.
8 Approximate the area under the function,, based on the given table values. Use a right hand approximation and 4 equal sub-intervals. Slide 33 / 175 A B C D E F G H I 9 Approximate the area under the function,, based on the given table values and intervals. Use a left hand approximation. Slide 34 / 175 Slide 35 / 175
Slide 36 / 175 Refresher on Summations: Slide 37 / 175 Refresher on Summations: Remind students how to calculate the summations, before the next slide where they will write their own to represent Riemann Sums. Slide 37 () / 175
Sigma Notation To represent Riemann Sums using sigma notation, we need to know the number of rectangles on the interval, and height of each rectangle. We will let represent each rectangle. Example: Use sigma notation to represent the area under the curve of on using 4 equal subintervals and left endpoints. Slide 38 / 175 Slide 38 () / 175 12 Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals? Slide 39 / 175 A B C D E
12 Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals? A C Slide 39 () / 175 B C D E 13 Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles? Slide 40 / 175 A B C D E 13 Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles? A B C D Slide 40 () / 175 D E
Slide 41 / 175 Trapezoid Approximation Return to Table of Contents Slide 41 () / 175 Teacher Notes Students may have already brought up the idea of using different shapes to approximate area; if not, you can Trapezoid bring Approximation it up now and ask for ideas to gain a closer approximation rather than using rectangles. Return to Table of Contents Trapezoidal Approximation Slide 42 / 175 Example: Approximate the area under the curve y = x 2 on [0,1] with using a trapezoidal approximation. Recall the area of a trapezoid:
Slide 42 () / 175 Simpson's Rule Slide 43 / 175 For future reference! We could make our approximation even closer if we used parabolas instead of lines as the tops of our intervals. This is called Simpson's Rule but this is not on the AP Calculus AB exam. 14 The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the height of each trapezoid? Slide 44 / 175
14 The area under the curve on [1,3] is approximated Students may have difficulty seeing the with 5 equal subintervals and trapezoids. What is the height of the trapezoid as the value, as height of each trapezoid? typically height is viewed vertically. It may help students to turn their papers sideways to calculate height and base length. Slide 44 () / 175 15 The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the area of the 5 th trapezoid? Slide 45 / 175 15 The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the area of the 5 th trapezoid? Slide 45 () / 175
16 The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the approximate area? Slide 46 / 175 16 The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the approximate area? Slide 46 () / 175 17 What is the approximate area under the curve on [0,9] using the given trapezoids? Slide 47 / 175
17 What is the approximate area under the curve on [0,9] using the given trapezoids? Remind students that intervals don't always have to be equal widths. Slide 47 () / 175 18 What is the approximate fuel consumed during the hour long flight using the trapezoids and given intervals? Slide 48 / 175 Time (minutes) Rate of Consumption (gal/min) 0 0 10 20 25 30 40 40 60 45 Slide 49 / 175 Area Under a Curve (The Definite Integral) Return to Table of Contents
What Do You Think? Slide 50 / 175 We have used rectangles and trapezoids to approximate the area under curves so far. What other techniques could we apply to gain a more accurate approximation of the area? Teacher Notes What Do You Think? Students may bring up various ideas about using different shapes. We have Guide used rectangles conversation and to trapezoids the idea that to approximate using the area more under and more curves rectangles so far. What will other techniques produce could a more we apply accurate to gain a more accurate approximation. of the area? Slide 50 () / 175 Infinite Rectangles Slide 51 / 175 If n is the number of rectangles used, and we allow that number to approach infinity, the width of each rectangle, or, will become infinitely small, which we denote. While the Greek symbol Sigma is a capital S for "sum", the German mathematician, Liebniz, chose to use the elongated "S" symbol for integrals in 1675, and it is still the symbol we use today.
The Definite Integral Slide 52 / 175 upper limit of integration differential (infinitely small ) integral sign lower limit of integration integrand (the function being integrated) The Definite Integral Slide 52 () / 175 upper limit of integration differential (infinitely small ) The expression is read as: integral sign Teacher Notes "The integral from a to b of f(x) dx." "The integral from a to b of f(x) with lower limit of integration respect to x." integrand (the function being integrated) or The Definite Integral Slide 53 / 175 If is continuous on [a,b] then the area under the curve is the integral of from a to b. Note: The integral represents the "net area" meaning all area above the x-axis minus any area below the x-axis.
Slide 54 / 175 19 Given the following: Slide 55 / 175 Find the value of the following integral, if possible. 19 Given the following: Slide 55 () / 175 Find the value of the following integral, if possible.
20 Given the following: Slide 56 / 175 Find the value of the following integral, if possible. 20 Given the following: Slide 56 () / 175 Find the value of the following integral, if possible. 21 Given the following: Slide 57 / 175 Find the value of the following integral, if possible.
21 Given the following: Slide 57 () / 175 Find the value of the following integral, Not possible; if possible. not given enough information 22 Given the following: Slide 58 / 175 Find the value of the following integral, if possible. 22 Given the following: Slide 58 () / 175 Find the value of the following integral, if possible.
23 Given the following: Slide 59 / 175 Find the value of the following integral, if possible. 23 Given the following: Slide 59 () / 175 Find the value of the following integral, Not possible; if possible. not given enough information 24 Given the following: Slide 60 / 175 Find the value of the following integral, if possible.
24 Given the following: Slide 60 () / 175 Find the value of the following integral, if possible. 25 Given the following: Slide 61 / 175 Find the value of the following integral, if possible. 25 Given the following: Slide 61 () / 175 Find the value of the following integral, Not possible; if possible. not enough information given
26 Given the following: Slide 62 / 175 Find the value of the following integral, if possible. 26 Given the following: Slide 62 () / 175 Find the value of the following integral, if possible. Evaluating Integrals Using Geometry Slide 63 / 175 Example: Using your knowledge of geometry, evaluate the following integral:
Evaluating Integrals Using Geometry Slide 63 () / 175 Example: Using your knowledge of geometry, evaluate the following integral: Students should recognize the equation for the semicircle. At this point, they will simply use the Area formula to evaluate the integral. 27 Evaluate: Slide 64 / 175 27 Evaluate: Slide 64 () / 175 Again, at this point students are simply using the geometric representation to calculate the area, as they haven't encountered antiderivatives yet.
28 Given the fact that use your knowledge of trig functions to evaluate: Slide 65 / 175 28 Given the fact that use your knowledge of trig functions to evaluate: Slide 65 () / 175 Slide 66 / 175 DISCUSSION: What does it mean when the area under the curve on a given interval equals zero?
Slide 66 () / 175 DISCUSSION: There is equal area above and below the What does it mean when the area x-axis under on the the curve given on a given interval equals zero? interval. Teacher Notes Slide 67 / 175 Antiderivatives & The Fundamental Theorem of Calculus Part II Return to Table of Contents Slide 68 / 175 What about other functions? In previous examples, we have either known the shape of the function to calculate the area, or information about the area was given to us. Now we will discover how to calculate the integral (area) for almost any function.
Recall: Let's take a look back at the example we did in the previous section: Slide 69 / 175 Let's imagine this representation is somebody running 5mph from 1 o'clock to 7 o'clock, it's simple to see the person traveled 30 miles. The area under the velocity function gives us the distance traveled. It was this notion that allowed mathematicians to discover the relationship between a function and it's derivative, and furthermore, a function's antiderivative. So, what exactly is an antiderivative? Slide 70 / 175 So, what exactly is an antiderivative? Allow students to discuss their ideas of an antiderivative and record all of their comments. Some common phrases/words may include: Teacher Notes "undo derivative" "go backwards" "reverse" Slide 70 () / 175 "find original function"
Antiderivative: Slide 71 / 175 Slide 71 () / 175 Slide 72 / 175
Slide 72 () / 175 Fundamental Theorem of Calculus, Part II Slide 73 / 175 Example: Evaluate the following integral: Slide 73 () / 175
Calculating Antiderivatives Slide 74 / 175 Calculating Antiderivatives Teacher Notes This slide is meant to help them with antiderivatives, but needs to be mathematically accurate as well. Since we are not delving into indefinite integrals quite yet, you may need to briefly discuss with students why there are no bounds for integration and why the +C constant value is written with the antiderivative. They will be introduced to this with indefinite integrals and will even revisit definite integrals again to see why it wasn't needed. Slide 74 () / 175 Fundamental Theorem of Calculus, Part II Slide 75 / 175 One thing to keep in mind is that it does not matter what variables are represented in your integral, as long as they match the variable you are integrating with respect to.
29 Evaluate: Slide 76 / 175 29 Evaluate: Slide 76 () / 175 30 Evaluate: Slide 77 / 175
30 Evaluate: Slide 77 () / 175 31 Evaluate: Slide 78 / 175 31 Evaluate: Slide 78 () / 175
32 Evaluate: Slide 79 / 175 32 Evaluate: Slide 79 () / 175 Slide 80 / 175
Slide 80 () / 175 34 Evaluate: Slide 81 / 175 A B C D E F 34 Evaluate: Slide 81 () / 175 A B D E C C F
Slide 82 / 175 The Fundamental Theorem of Calculus, Part I Return to Table of Contents Fundamental Theorem of Calculus Slide 83 / 175 Most mathematicians consider The Fundamental Theorem of Calculus as the most important discovery in the history of mathematics. This relationship between differentiation and integration provided a critical connection between the two fields which first appeared unrelated. Slide 84 / 175
Slide 84 () / 175 Fundamental Theorem of Calculus, Part I Slide 85 / 175 If can evaluate :, then using our previous knowledge of integration, we Slide 85 () / 175
Fundamental Theorem of Calculus, Part I Slide 86 / 175 Now, taking this one step further... Let's calculate the derivative of f(x). Fundamental Theorem of Calculus, Part I Slide 87 / 175 Putting it all together, we calculated the following: Can you make any observations about methods to get from the first equation to the last and omitting the middle step? Fundamental Theorem of Calculus, Part I Slide 87 () / 175 Putting it all together, we calculated Have students the following: discuss their thoughts and ideas about this process. It is not always clear to all students straight away, so ask leading questions about what the function and derivative have in common, etc. Teacher Notes Can you make any observations about [This object methods is a pull tab] to get from the first equation to the last and omitting the middle step?
Fundamental Theorem of Calculus, Part I Slide 88 / 175 Our work on previous slides has led us to the discovery of the Fundamental Theorem of Calculus, Part I which states: If is a continuous function on, then Fundamental Theorem of Calculus, Part I Slide 89 / 175 In common terms, if taking the derivative of an integral, evaluated from a constant to x, you can simply replace the variable in the integral with x for your derivative. Note: This only applies when the lower limit of integration is constant and the upper limit is x. We will soon discuss how to evaluate if it is something other than x. FTC (Part I) Slide 90 / 175 Let's look at how this theorem works with another function: Find the derivative of:
FTC (Part I) Slide 90 () / 175 It is worth mentioning to Let's look at how this theorem students works with that another an added function: benefit of using the FTC part I is that it allows us to differentiate Find the derivative of: integrals for which we do not know the antiderivative of the integrand. Teacher Notes FTC (Part I) Slide 91 / 175 Example: Given Find. FTC (Part I) Slide 91 () / 175 Example: Given Find.
FTC (Part I) Slide 92 / 175 Example: Given Find. FTC (Part I) Slide 92 () / 175 Example: Given Find. Special Circumstances Now let's discuss special circumstances of the FTC: What do you notice is different about the following example? Given Find. Slide 93 / 175
Special Circumstances Students should recognize the bounds of Now let's discuss special circumstances of the FTC: integration are flipped. They can use the What do you notice is properties different of about integrals the following to rewrite example? the Given integral Find and. apply the FTC. Slide 93 () / 175 Special Circumstances Another special circumstance of the FTC: What do you notice is different about the following example? Given Find. Slide 94 / 175 Special Circumstances Students should recognize the upper limit Another special circumstance of the FTC: of integration is not just x. We must apply What do you notice a is form different of the about Chain the Rule following with the example? FTC. Given Find. Extra step: Slide 94 () / 175
Special Circumstances One more special circumstance of the FTC: What do you notice is different about the following example? Given Find. Slide 95 / 175 Slide 95 () / 175 35 Find Slide 96 / 175 A B C D E
35 Find Slide 96 () / 175 A B C D E C 36 Find Slide 97 / 175 A B C D E 36 Find Slide 97 () / 175 A B C D E B
37 Find Slide 98 / 175 A B C D E 37 Find Slide 98 () / 175 A B C D E D 38 If, find Slide 99 / 175 *From the 1976 AP Calculus AB Exam
38 If, find Slide 99 () / 175 *From the 1976 AP Calculus AB Exam 39 Find Slide 100 / 175 A B C D E 39 Find Slide 100 () / 175 A B C D E A
Slide 101 / 175 Slide 101 () / 175 41 Slide 102 / 175
41 Slide 102 () / 175 Slide 103 / 175 Slide 104 / 175 Average Value & Mean Value Theorem for Integrals Return to Table of Contents
Average Value Slide 105 / 175 Recall the graph below which we used at the beginning of the unit regarding a non-constant speed. This section will allow us to calculate the average value (in this case, the average velocity) on a given interval. If we are given a function, we can then apply our knowledge of integrals to calculate this value. v(t) (mph) 50 mph 30 mph 5 hrs t (hours) Average Value Slide 106 / 175 Imagine all of the area under the given curve is transformed into a rectangle. This rectangle has the same base length as the interval. The height is the average value the functions takes on within that interval. The star represents the average value. The Average Value of f(x) is the height of the rectangle with base (b-a) and whose area equals the area under the graph of f(x) between x=a and x=b. Average Value Slide 107 / 175 If is a continuous function on. the average value of the function is:
Slide 108 / 175 Slide 108 () / 175 Average Value Slide 109 / 175 A graphical representation of our answer from the previous example:
Average Value Slide 110 / 175 Another example: Find the average value of the function over the given interval. Slide 110 () / 175 Average Value Slide 111 / 175 Note: The average value of a function is not found by averaging the 2 y-values of the interval boundaries.
Average Value This is an important idea to emphasize Note: The average to students. value Review of a function the last is example not found by averaging the to make 2 y-values the point of the clear. interval boundaries. Teacher Notes It is a common misconception to just calculate f(0) and f(4) and average them, which in this case would equal 1. When applying the formula correctly we found the average value to be 1.333. Slide 111 () / 175 Slide 112 / 175 Slide 112 () / 175
Slide 113 / 175 Slide 113 () / 175 43 Find the average value of the function on the given interval. Slide 114 / 175 A B C D E F
43 Find the average value of the function on the given interval. Slide 114 () / 175 A D C B E C F Slide 115 / 175 Slide 115 () / 175
Mean Value Theorem for Integrals Slide 116 / 175 Much like the MVT for Derivatives told us at which value, c, the slope was equal to the average slope; the Mean Value Theorem for Integrals will tell us at which value, c, the function reaches it's average value. Mean Value Theorem for Integrals Slide 117 / 175 Let's just say we already calculated the average value for our function, and found it to be 30mph. v(t) (mph) 50 mph 30 mph 5 hrs t (hours) The Mean Value Theorem for Integrals states that at at least one point, c, the function must take on it's average value. What does that mean for our example above? Mean Value Theorem for Integrals Slide 117 () / 175 Let's just say we already calculated the average value for our function, and found it to be 30mph. Students should discuss and conclude that although the car v(t) (mph) was not travelling at 30mph the entire trip, there must have 50 mph been at least one point on the 30 mph trip that it was traveling at that speed. In this case, by observing the graph, 2 points. 5 hrs t (hours) Teacher Notes The Mean Value Theorem for Integrals states that at at least one point, c, the function must take on it's average value. What does that mean for our example above?
Mean Value Theorem for Integrals Slide 118 / 175 If f(x) is a continuous function on [a,b], then at some point, c, where a<c<b Slide 119 / 175 Slide 119 () / 175
45 Find the value(s) of c that satisfy the MVT for integrals. Slide 120 / 175 A B C D E F 45 Find the value(s) of c that satisfy the MVT for integrals. Slide 120 () / 175 A B D E E C F Slide 121 / 175
Slide 121 () / 175 Slide 122 / 175 Slide 122 () / 175
48 Find the value(s) of c that satisfy the Mean Value Theorem for integrals. Slide 123 / 175 Slide 123 () / 175 Slide 124 / 175 Indefinite Integrals Return to Table of Contents
Indefinite Integrals Slide 125 / 175 So far, we have been calculating definite integrals, meaning our integral had bounds on both sides. Next, we will consider what happens when our integrals do not contain upper or lower limits, also known as indefinite integrals. Slide 126 / 175 Slide 126 () / 175
Reflect Slide 127 / 175 Talk, in teams, about what you noticed about the functions written in each box. Indefinite Integrals = Antiderivatives Slide 128 / 175 When we are given integrals without upper and lower limits, the question is really asking us to give the antiderivative. The only catch is that we MUST include the +C constant with each antiderivative. This represents that technically any constant could be added to the original function. In the next unit, we will see that sometimes additional information (like an initial condition) is provided in order to find out exactly what the original function was. If not, it is imperative to include the +C. Indefinite Integrals Slide 129 / 175
Example Slide 130 / 175 Evaluate: Example Slide 130 () / 175 Evaluate: Example Slide 131 / 175 Evaluate:
Example Slide 131 () / 175 Evaluate: How Important is the Constant? Slide 132 / 175 It might seem like a trivial piece, but when evaluating indefinite integrals, it is extremely critical to include the +C value. In fact, when questions arise on the free response portion of the AP Exam, 1 point is often awarded just for including the +C. Keep in mind, only 9 points area available on each free response question. Including the constant of integration not only demonstrates that you have a good understanding of the behavior of antiderivatives, but also allows you to take questions further as we will see with differential equations in the next unit. Slide 133 / 175
Slide 133 () / 175 Antiderivatives Involving Exponential and Natural Log Functions Slide 134 / 175 Antiderivatives Involving Exponential and Natural Log Functions Slide 134 () / 175 Teacher Notes It's important to point out the necessity of the absolute value bars in ln x. No matter what number e is raised to, the result will be positive.
49 Slide 135 / 175 49 Slide 135 () / 175 Note: These response questions intentionally avoid using multiple choice to allow students the opportunity to remember to include the +C with their answer. 50 Slide 136 / 175
50 Slide 136 () / 175 51 Slide 137 / 175 51 Slide 137 () / 175
52 Slide 138 / 175 52 Slide 138 () / 175 53 Slide 139 / 175
53 Slide 139 () / 175 54 Slide 140 / 175 54 Slide 140 () / 175
55 Slide 141 / 175 55 Slide 141 () / 175 Recap: Definite vs. Indefinite Integrals Slide 142 / 175 Turn to a partner to discuss the similarities and differences of definite and indefinite integrals.
Slide 143 / 175 U-Substitution Return to Table of Contents U-Substitution Slide 144 / 175 Recall when we were asked to find the derivative of a composite function. We had to utilize the Chain Rule to take the derivative correctly. U-Substitution is a similar tool used to find the antiderivative of more complex functions, and essentially the "undo" of the Chain Rule. U-Substitution Slide 145 / 175 Let's start with an example. Evaluate: Notice, upon first glance this looks like a fairly complex integral. But, if we let We can then find the differential: This allows us to rewrite the integral in terms of u, to make the integration easier. Notice how much less intimidating this integral is!
U-Substitution Slide 146 / 175 Now, let's finish evaluating the integral. DON'T FORGET! Substitute your expression back in for u. Steps for U-Substitution Slide 147 / 175 1. Choose your value for u. 2. Find the differential (take derivative & solve for du). 3. Make the substitution into original problem. 4. Integrate as usual. 5. Substitute back in for u. Deciding Values for U Slide 148 / 175 In our previous example, the value for u was given; however, this won't always be the case. So, how do we decide or choose the correct value for u? The best advice is to look for an expression in the integral for which you also see that expression's derivative.
More Tips for Choosing Your "U" Slide 149 / 175 DOs Good choices for u usually include: An expression raised to a power An expression in a denominator An expression as an "inside" function of a composition DONTs Avoid choosing a value for u which is too complex, at the same time make sure it's not too simple (i.e. don't let u=x) Slide 150 / 175 Slide 150 () / 175
Slide 151 / 175 Slide 151 () / 175 Slide 152 / 175
Slide 152 () / 175 Slide 153 / 175 Slide 153 () / 175
Slide 154 / 175 Slide 154 () / 175 New Circumstance Slide 155 / 175 Consider the following example: If we let: Then, What do you notice?
New Circumstance Slide 155 () / 175 Consider the following example: Allow students to discuss their findings in pairs or groups. Students If we let: should recognize that the exact equation we have for du is not found Then, in the original integral. It's different by a factor of 2. Teacher Notes What do you notice? Fixing the Problem Slide 156 / 175 Evaluate: In order to make our substitution this time, we must make one additional step. Fixing the Problem Slide 156 () / 175 Divide by 2 on both sides: Evaluate: So, our new integral becomes: (Recall the properties of integrals allow us to bring the constant outside the integral)
One More Situation to Consider Consider the following example: Slide 157 / 175 Slide 157 () / 175 61 Evaluate using u-substitution (if needed): Slide 158 / 175
61 Evaluate using u-substitution (if needed): Slide 158 () / 175 Make sure students understand the importance of adding the +C!!! 62 Evaluate using u-substitution (if needed): Slide 159 / 175 A B C D E 62 Evaluate using u-substitution (if needed): Slide 159 () / 175 A B C A D E
63 Evaluate using u-substitution (if needed): Slide 160 / 175 63 Evaluate using u-substitution (if needed): Slide 160 () / 175 64 Evaluate using u-substitution (if needed): Slide 161 / 175 A B C D E
64 Evaluate using u-substitution (if needed): Slide 161 () / 175 A B B C D E 65 Evaluate using u-substitution (if needed): Slide 162 / 175 A B C D E 65 Evaluate using u-substitution (if needed): A B C By this point, students are in the habit of using u-substitution. However, this question can be approached by simply distributing first, and then integrating like normal. D Slide 162 () / 175 D E
66 Evaluate using u-substitution (if needed): Slide 163 / 175 66 Evaluate using u-substitution (if needed): Slide 163 () / 175 67 Evaluate using u-substitution (if needed): CHALLENGE Slide 164 / 175
Slide 164 () / 175 U-Substitution with Definite Integrals Slide 165 / 175 Definite integrals may sometimes require u-substitution as well, but it is important to take extra caution when dealing with the limits of integration. Let's look at an example and then address the issue with the limits. Note: When we are given the original problem, all expressions and values are in terms of the variable x, including the bounds of integration. U-Substitution with Definite Integrals Slide 166 / 175 From this point, you have 2 options: 1. Integrate, substitute the expression with x back in and use original bounds. OR 2. Change bounds in terms of u, integrate and use the new bounds to evaluate. We will finish this example showing both methods to compare.
U-Substitution with Definite Integrals Slide 167 / 175 Option #1: Integrate, substitute the expression with x back in and use original bounds. Slide 167 () / 175 U-Substitution with Definite Integrals Slide 168 / 175 Option #2: Change bounds in terms of u, integrate and use the new bounds to evaluate.
Slide 168 () / 175 68 Which values correspond to the correct bounds of integration in terms of u? Slide 169 / 175 A B C D 68 Which values correspond to the correct bounds of integration in terms of u? A B C D A Slide 169 () / 175
69 Which values correspond to the correct bounds of integration in terms of u? Slide 170 / 175 A B C D 69 Which values correspond to the correct bounds of integration in terms of u? Slide 170 () / 175 A B C D C 70 Given that, which of the following answers is equivalent to? Slide 171 / 175 A B C D E
70 Given that, which of the following answers is equivalent to? A D Slide 171 () / 175 B C D E Slide 172 / 175 Slide 172 () / 175
72 Slide 173 / 175 A B C D E 72 Slide 173 () / 175 A B C D E B 73 Evaluate: Slide 174 / 175
73 Evaluate: Slide 174 () / 175 74 Evaluate: Slide 175 / 175 Slide 175 () / 175