AP Calculus AB Integration 2015 11 24 www.njctl.org Table of Contents click on the topic to go to that section 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 1
Integration 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. Riemann Sums Return to Table of Contents 2
Area of Curved Shapes 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 Consider the following velocity graph: v(t) (mph) 30 mph t (hours) 5 hrs How far did the person drive? 3
Non Constant Speed 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. 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. 4
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. Teacher Notes RAM Rectangular Approximation Method Rectangular Approximation Method is a way to estimate area by drawing rectangles from the x axis up to the curve. 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) 5
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 left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. 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 left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain. First, determine the width of each rectangle/ interval if there are 4 total. Second, draw the rectangles in with the left hand corner touching the function. 6
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 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 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. 7
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 right endpoints (RRAM). Is this approximation an overestimate or an underestimate? 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). 8
Riemann Sums MRAM *NOTE: LRAM + RRAM 2 Teacher Notes When approximating the area under the curve y=3x+2 on [1,4] using four rectangles, how wide should each sub interval be? 9
Approximate the area under y=3x+2 on [1,4] using six rectangles and LRAM. Find the area under the curve on [ 3,2] using five sub intervals and RRAM. 10
1 When approximating the area under on [1,3] using four rectangles and MRAM, what x value should be used to calculate the height of the third rectangle? A D G B C E F H I 2 Which of the following correctly shows the right hand approximation of the area under on [1,7] using 4 sub intervals? A B C D E 11
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. Teacher Notes Using the symbols above, can you create a mathematical relationship between all 4 of them? Calculate for the left hand approximation of the area under if and. 12
3 What is the value of n when approximating the area under if, and? A B C D E F Riemann Sums with Tables 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. 13
Riemann Sums with Tables Example: Approximate the area under the curve, f(x), on [ 2,4] using right endpoints and n=3. Riemann Sums with Tables Example: Approximate the area under the curve, f(x), on [ 2,4] using right endpoints and n=3. RRAM= 14
Riemann Sums with Tables 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 Example: Using the subintervals in the table, approximate the area under using a left hand approximation. 15
Riemann Sums with Tables Example: Using the subintervals in the table, approximate the area under using a right hand approximation. 4 Approximate the area under the function,, based on the given table values. Use a right hand approximation and 4 equal sub intervals. A B C D E F G H I 16
Approximate the area under the function,, based on the given table values and intervals. Use a left hand approximation. 5 It is possible to use a mid point approximation for the area under the following curve using the table data if. True False 17
6 It is possible to use a mid point approximation for the area under the following curve using the table data if. True False Refresher on Summations: Remind students how to calculate the summations, before the next slide where they will write their own to represent Riemann Sums. [This object is a pull tab] 18
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. 7 Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals? A B C D E 19
8 Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles? A B C D E Trapezoid Approximation Teacher Notes Return to Table of Contents 20
Trapezoidal Approximation Example: Approximate the area under the curve y = x 2 on [0,1] with using a trapezoidal approximation. Recall the area of a trapezoid: Simpson's Rule 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. 21
The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the height of each trapezoid? 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? 22
The area under the curve on [1,3] is approximated with 5 equal subintervals and trapezoids. What is the approximate area? [This object is a pull tab] What is the approximate area under the curve on [0,9] using the given trapezoids? 23
What is the approximate fuel consumed during the hour long flight using the trapezoids and given intervals? Time (minutes) Rate of Consumption (gal/min) 0 0 10 20 25 30 40 40 60 45 Area Under a Curve (The Definite Integral) Return to Table of Contents 24
What Do You Think? 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 Infinite Rectangles 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. 25
The Definite Integral upper limit of integration differential (infinitely small ) integral sign lower limit of integration integrand (the function being integrated) Teacher Notes The Definite Integral 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. 26
Properties of Integrals where k is a constant Given the following: Find the value of the following integral, if possible. 27
Given the following: Find the value of the following integral, if possible. Given the following: Find the value of the following integral, if possible. 28
Given the following: Find the value of the following integral, if possible. Given the following: Find the value of the following integral, if possible. 29
Given the following: Find the value of the following integral, if possible. Given the following: Find the value of the following integral, if possible. 30
Given the following: Find the value of the following integral, if possible. Evaluating Integrals Using Geometry Example: Using your knowledge of geometry, evaluate the following integral: 31
Evaluate: Given the fact that use your knowledge of trig functions to evaluate: 32
DISCUSSION: What does it mean when the area under the curve on a given interval equals zero? Teacher Notes Antiderivatives & The Fundamental Theorem of Calculus Part II Return to Table of Contents 33
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: 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. 34
So, what exactly is an antiderivative? Teacher Notes Antiderivative: Teacher Notes 35
Fundamental Theorem of Calculus, Part II Antiderivatives allow us to evaluate integrals, using the Fundamental Theorem of Calculus (part II). The theorem states: If is continuous at every point on and if is the antiderivative of on then Teacher Notes Fundamental Theorem of Calculus, Part II Example: Evaluate the following integral: 36
Calculating Antiderivatives Teacher Notes Fundamental Theorem of Calculus, Part II 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. 37
Evaluate: Evaluate: 38
Evaluate: Evaluate: 39
9 Evaluate: A B C D E F 10 Evaluate: A B C D E F 40
The Fundamental Theorem of Calculus, Part I Return to Table of Contents Fundamental Theorem of Calculus 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. 41
Fundamental Theorem of Calculus, Part I Consider the following graph: What does represent? Fundamental Theorem of Calculus, Part I If, then using our previous knowledge of integration, we can evaluate : 42
Fundamental Theorem of Calculus, Part I Now, taking this one step further... Let's calculate the derivative of f(x). Fundamental Theorem of Calculus, Part I Putting it all together, we calculated the following: Teacher Notes Can you make any observations about methods to get from the first equation to the last and omitting the middle step? 43
Fundamental Theorem of Calculus, Part I 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 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. 44
FTC (Part I) Let's look at how this theorem works with another function: Find the derivative of: Teacher Notes FTC (Part I) Example: Given Find. 45
FTC (Part I) 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. 46
Special Circumstances Another special circumstance of the FTC: What do you notice is different about the following example? Given Find. Special Circumstances One more special circumstance of the FTC: What do you notice is different about the following example? Given Find. 47
11 Find A B C D E 12 Find A B C D E 48
13 Find A B C D E If, find *From the 1976 AP Calculus AB Exam 49
14 Find A B C D E 15 A B C D E *From the 2003 AP Calculus AB Exam 50
Recap: Comparing the Fundamental Theorem of Calculus Part I and II FTC Part I: Allows us to find the derivative of definite integrals FTC Part II: Allows us to evaluate definite integrals 51
Average Value & Mean Value Theorem for Integrals Return to Table of Contents v(t) (mph) 50 mph 30 mph Average Value 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. 5 hrs t (hours) 52
Average Value 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 If is a continuous function on. the average value of the function is: 53
Average Value Let's try an example: Find the average value of the function over the given interval. Average Value A graphical representation of our answer from the previous example: 54
Average Value Another example: Find the average value of the function over the given interval. Average Value Note: The average value of a function is not found by averaging the 2 y values of the interval boundaries. Teacher Notes 55
Average Value What happens if we are asked to take the average value of a piecewise defined function? If the interval falls over both pieces, just split the integral into pieces as well. Find the average value of g(x) over the given interval: 16 Find the average value of the function on the given interval. A D B C E F 56
17 Find the average value of the function on the given interval. A B C D E F Find the average value of the function on the given interval. 57
Mean Value Theorem for Integrals 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 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? Teacher Notes 58
Mean Value Theorem for Integrals If f(x) is a continuous function on [a,b], then at some point, c, where a<c<b Let's try an example: MVT for Integrals Find the value(s) of c that satisfy the Mean Value Theorem for Integrals. 59
18 Find the value(s) of c that satisfy the MVT for integrals. A B C D E F 19 Find the value(s) of c that satisfy the MVT for integrals. A D B E C F 60
20 Find the value(s) c that satisfy the MVT for integrals. A B D E C F Find the value(s) of c that satisfy the Mean Value Theorem for integrals. 61
Indefinite Integrals Return to Table of Contents Indefinite Integrals 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. 62
Activity: List any possible functions that have the following derivatives: Teacher Notes Reflect Talk, in teams, about what you noticed about the functions written in each box. 63
Indefinite Integrals = Antiderivatives 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 64
Example Evaluate: Example Evaluate: 65
How Important is the Constant? 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. Why Didn't we Include +C for Definite Integrals? When we calculated definite integrals in previous sections, there was no need to include the constant, C, in our answers. Let's take a look at a basic example to see why it was unnecessary. Evaluate: Teacher Notes What happens? 66
Antiderivatives Involving Exponential and Natural Log Functions Teacher Notes 67
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Recap: Definite vs. Indefinite Integrals Turn to a partner to discuss the similarities and differences of definite and indefinite integrals. U Substitution Return to Table of Contents 71
U Substitution 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 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! 72
U Substitution Now, let's finish evaluating the integral. DON'T FORGET! Substitute your expression back in for u. Steps for U Substitution 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. 73
Deciding Values for U 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" 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) 74
21 Which expression represents the correct choice for u in order to evaluate the integral? A B C D E 22 Which expression represents the correct choice for u in order to evaluate the integral? A B C D E 75
23 Which expression represents the correct choice for u in order to evaluate the integral? A B C D E 24 Which expression represents the correct choice for u in order to evaluate the integral? A B C D E 76
25 Which expression represents the correct choice for u in order to evaluate the integral? A B C D E New Circumstance Consider the following example: If we let: Then, Teacher Notes What do you notice? 77
Fixing the Problem Evaluate: One More Situation to Consider Consider the following example: Teacher Notes 78
Evaluate using u substitution (if needed): 26 Evaluate using u substitution (if needed): A B C D E 79
Evaluate using u substitution (if needed): 27 Evaluate using u substitution (if needed): A B C D E 80
28 Evaluate using u substitution (if needed): A B C D E Evaluate using u substitution (if needed): 81
Evaluate using u substitution (if needed): CHALLENGE U Substitution with Definite Integrals 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. 82
U Substitution with Definite Integrals 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 Option #1: Integrate, substitute the expression with x back in and use original bounds. 83
U Substitution with Definite Integrals Option #2: Change bounds in terms of u, integrate and use the new bounds to evaluate. 29 Which values correspond to the correct bounds of integration in terms of u? A B C D 84
30 Which values correspond to the correct bounds of integration in terms of u? A B C D 31 Given that, which of the following answers is equivalent to? A B C D E 85
32 A B C D E 33 A B C D E 86
Evaluate: Evaluate: 87