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Slide 2 / 175 AP Calculus AB Integration 2015-11-24 www.njctl.org
Slide 3 / 175 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
Slide 4 / 175 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.
Slide 5 / 175 Riemann Sums Return to Table of Contents
Slide 6 / 175 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.
Slide 7 / 175 Distance Using Graphs Consider the following velocity graph: v(t) (mph) 30 mph 5 hrs t (hours) How far did the person drive?
Slide 8 / 175 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.
Slide 9 / 175 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.
Slide 10 / 175 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.
Slide 11 / 175 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)
Slide 12 / 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 left endpoints (LRAM). Is this approximation an overestimate or an underestimate? Explain.
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Slide 14 / 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 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.
Slide 15 / 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 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.
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 right endpoints (RRAM). Is this approximation an overestimate or an underestimate?
Slide 17 / 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 18 / 175 Riemann Sums *NOTE: MRAM LRAM + RRAM 2
Slide 19 / 175 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 20 / 175 2 Approximate the area under y=3x+2 on [1,4] using six rectangles and LRAM.
Slide 21 / 175 3 Find the area under the curve on [-3,2] using five sub-intervals and RRAM.
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Slide 24 / 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. Using the symbols above, can you create a mathematical relationship between all 4 of them?
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Slide 27 / 175 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.
Slide 28 / 175 Riemann Sums with Tables Example: Approximate the area under the curve, f(x), on [-2,4] using right endpoints and n=3.
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Slide 30 / 175 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.
Slide 31 / 175 Riemann Sums with Tables Example: Using the subintervals in the table, approximate the area under using a left hand approximation. Answer
Slide 32 / 175 Riemann Sums with Tables Example: Using the subintervals in the table, approximate the area under using a right hand approximation. Answer
8 Slide 33 / 175 Approximate the area under the function,, based on the given table values. Use a right hand approximation and 4 equal sub-intervals. Answer A B C D E F G H I
Slide 34 / 175 9 Approximate the area under the function,, based on the given table values and intervals. Use a left hand approximation. Answer
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Slide 37 / 175 Refresher on Summations:
Slide 38 / 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 39 / 175 12 Which of the following represents the approximate area under the curve on using midpoints and 3 equal subintervals? A B C D E
Slide 40 / 175 13 Which of the following represents the approximate area under the curve on using right endpoints and 6 rectangles? A B C D E
Slide 41 / 175 Trapezoid Approximation Return to Table of Contents
Slide 42 / 175 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:
Slide 43 / 175 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.
Slide 44 / 175 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 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 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 47 / 175 17 What is the approximate area under the curve on [0,9] using the given trapezoids?
Slide 48 / 175 18 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 Answer 10 20 25 30 40 40 60 45
Slide 49 / 175 Area Under a Curve (The Definite Integral) Return to Table of Contents
Slide 50 / 175 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?
Slide 51 / 175 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.
Slide 52 / 175 The Definite Integral upper limit of integration differential (infinitely small ) integral sign lower limit of integration integrand (the function being integrated)
Slide 53 / 175 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.
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Slide 55 / 175 19 Given the following: Find the value of the following integral, if possible.
Slide 56 / 175 20 Given the following: Find the value of the following integral, if possible.
Slide 57 / 175 21 Given the following: Find the value of the following integral, if possible.
Slide 58 / 175 22 Given the following: Find the value of the following integral, if possible.
Slide 59 / 175 23 Given the following: Find the value of the following integral, if possible.
Slide 60 / 175 24 Given the following: Find the value of the following integral, if possible.
Slide 61 / 175 25 Given the following: Find the value of the following integral, if possible.
Slide 62 / 175 26 Given the following: Find the value of the following integral, if possible.
Slide 63 / 175 Evaluating Integrals Using Geometry Example: Using your knowledge of geometry, evaluate the following integral:
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Slide 65 / 175 28 Given the fact that use your knowledge of trig functions to evaluate:
Slide 66 / 175 DISCUSSION: What does it mean when the area under the curve on a given interval equals zero?
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.
Slide 69 / 175 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.
So, what exactly is an antiderivative? Slide 70 / 175
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Slide 73 / 175 Fundamental Theorem of Calculus, Part II Example: Evaluate the following integral:
Slide 74 / 175 Calculating Antiderivatives
Slide 75 / 175 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.
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Slide 81 / 175 34 Evaluate: A B C D E F
Slide 82 / 175 The Fundamental Theorem of Calculus, Part I Return to Table of Contents
Slide 83 / 175 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.
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Slide 85 / 175 Fundamental Theorem of Calculus, Part I If, then using our previous knowledge of integration, we can evaluate :
Slide 86 / 175 Fundamental Theorem of Calculus, Part I Now, taking this one step further... Let's calculate the derivative of f(x).
Slide 87 / 175 Fundamental Theorem of Calculus, Part I 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?
Slide 88 / 175 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
Slide 89 / 175 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.
Slide 90 / 175 FTC (Part I) Let's look at how this theorem works with another function: Find the derivative of:
Slide 91 / 175 FTC (Part I) Example: Given Find.
Slide 92 / 175 FTC (Part I) Example: Given Find.
Slide 93 / 175 Special Circumstances Now let's discuss special circumstances of the FTC: What do you notice is different about the following example? Given Find.
Slide 94 / 175 Special Circumstances Another special circumstance of the FTC: What do you notice is different about the following example? Given Find.
Slide 95 / 175 Special Circumstances One more special circumstance of the FTC: What do you notice is different about the following example? Given Find.
Slide 96 / 175 35 Find A B C D E
Slide 97 / 175 36 Find A B C D E
Slide 98 / 175 37 Find A B C D E
Slide 99 / 175 38 If, find *From the 1976 AP Calculus AB Exam
Slide 100 / 175 39 Find A B C D E
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Slide 104 / 175 Average Value & Mean Value Theorem for Integrals Return to Table of Contents
v(t) (mph) 50 mph 30 mph Slide 105 / 175 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)
Slide 106 / 175 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.
Slide 107 / 175 Average Value If is a continuous function on. the average value of the function is:
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Slide 109 / 175 Average Value A graphical representation of our answer from the previous example:
Slide 110 / 175 Average Value Another example: Find the average value of the function over the given interval.
Slide 111 / 175 Average Value Note: The average value of a function is not found by averaging the 2 y-values of the interval boundaries.
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Slide 114 / 175 43 Find the average value of the function on the given interval. A B C D E F
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Slide 116 / 175 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.
Slide 117 / 175 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?
Slide 118 / 175 Mean Value Theorem for Integrals If f(x) is a continuous function on [a,b], then at some point, c, where a<c<b
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Slide 120 / 175 45 Find the value(s) of c that satisfy the MVT for integrals. A B C D E F
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Slide 123 / 175 48 Find the value(s) of c that satisfy the Mean Value Theorem for integrals.
Slide 124 / 175 Indefinite Integrals Return to Table of Contents
Slide 125 / 175 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.
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Slide 127 / 175 Reflect Talk, in teams, about what you noticed about the functions written in each box.
Slide 128 / 175 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.
Slide 129 / 175 Indefinite Integrals
Slide 130 / 175 Example Evaluate:
Slide 131 / 175 Example Evaluate:
Slide 132 / 175 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.
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Slide 134 / 175 Antiderivatives Involving Exponential and Natural Log Functions
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Slide 142 / 175 Recap: Definite vs. Indefinite Integrals 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
Slide 144 / 175 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.
Slide 145 / 175 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!
Slide 146 / 175 U-Substitution Now, let's finish evaluating the integral. DON'T FORGET! Substitute your expression back in for u.
Slide 147 / 175 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.
Slide 148 / 175 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.
Slide 149 / 175 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)
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Slide 155 / 175 New Circumstance Consider the following example: If we let: Then, What do you notice?
Slide 156 / 175 Fixing the Problem Evaluate:
Slide 157 / 175 One More Situation to Consider Consider the following example:
Slide 158 / 175 61 Evaluate using u-substitution (if needed):
Slide 159 / 175 62 Evaluate using u-substitution (if needed): A B C D E
Slide 160 / 175 63 Evaluate using u-substitution (if needed):
Slide 161 / 175 64 Evaluate using u-substitution (if needed): A B C D E
Slide 162 / 175 65 Evaluate using u-substitution (if needed): A B C D E
Slide 163 / 175 66 Evaluate using u-substitution (if needed):
Slide 164 / 175 67 Evaluate using u-substitution (if needed): CHALLENGE
Slide 165 / 175 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.
Slide 166 / 175 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.
Slide 167 / 175 U-Substitution with Definite Integrals Option #1: Integrate, substitute the expression with x back in and use original bounds.
Slide 168 / 175 U-Substitution with Definite Integrals Option #2: Change bounds in terms of u, integrate and use the new bounds to evaluate.
Slide 169 / 175 68 Which values correspond to the correct bounds of integration in terms of u? A B C D
Slide 170 / 175 69 Which values correspond to the correct bounds of integration in terms of u? A B C D
Slide 171 / 175 70 Given that, which of the following answers is equivalent to? A B C D E
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Slide 173 / 175 72 A B C D E
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