Slide 1 / 163 Slide 2 / 163 Pre-alculus Intro to Integrals 2015-03-24 www.njctl.org Slide 3 / 163 Table of ontents Riemann Sums Trapezoid Rule ccumulation Function & efinite Integrals Fundamental Theorem of alculus rea etween urves Volume: Shell Method click on the topic to go to that section Slide 4 / 163 Riemann Sums Return to Table of ontents Slide 5 / 163 Slide 6 / 163 onsider the following velocity graph: 30 mph ut we seldom travel at a constant speed. 50 mph 5 hrs How far did the person drive? The area under the velocity graph is the total distance traveled. Integration is used to find the area. 5 hrs The area under this graph is still the distance traveled but we need more than multiplication to find it.
Slide 7 / 163 Slide 8 / 163 George Riemann (Re-mon) studied making these curves into a series of rectangles. Riemann Sums, or Rectangular pproximation Method (RM), is calculated by drawing rectangles from the x-axis up to the curve. The question is: What part of the "top" of the rectangle should be used to determine the height of the rectangle? So the area under the curve would be the sum of areas of the rectangles, this is called Riemann Sums. The right hand corner. (RRM) The left hand corner. (LRM) The middle. (MRM) Slide 9 / 163 Slide 10 / 163 Example: Find the area between y = x 2, the x-axis, and [0,1] using Riemann Sums and 4 partitions. LRM Example: Find the area between y = x 2, the x-axis, and [0,1] using Riemann Sums and 4 partitions. RRM 0 1/4 1/2 3/4 1 Found the width of the rectangle: (b-a)/n= (1-0)/4 = 1/4 0 1/4 1/2 3/4 1 Is this approximation an overestimate or an underestimate? Is this approximation an overestimate or an underestimate? Slide 11 / 163 Slide 12 / 163 Example: Find the area between y = x 2, the x-axis, and [0,1] using Riemann Sums and 4 partitions. MRM *NOTE: MRM LRM + RRM 2 0 1/4 1/2 3/4 1 This value falls between LRM and RRM.
Slide 13 / 163 Slide 14 / 163 1 When finding the area between and the x-axis [1,3] using four partitions, how wide should each interval be? Slide 15 / 163 Slide 16 / 163 2 Find the area between and the x-axis [1,3] using four partitions and LRM. 3 Find the area between and the x-axis [1,3] using four partitions and RRM. Slide 17 / 163 Slide 18 / 163 4 When finding the area between and the x-axis [1,3] using four partitions and MRM, when in the third rectangle, what x should be used to find the height? 5 Find the area between and the x-axis [1,3] using four partitions and MRM.
Slide 19 / 163 Slide 20 / 163 We can write the four areas using Selected Rules for Sigma where a k is the area of the k th rectangle. It is just another way of writing what we just did. # is the Greek letter sigma and stands for the summation of all the terms evaluated at starting with the bottom number and going through to the top. Slide 21 / 163 Slide 22 / 163 1 st n integers: Equivalent Formulas 6 1 st n squares: 1 st n cubes: Slide 23 / 163 Slide 24 / 163 7
Slide 25 / 163 Slide 26 / 163 Trapezoid Rule Return to Table of ontents Slide 27 / 163 Slide 28 / 163 Example: Find the area y = x 2 and the x-axis [0,1] using 4 partitions. Example: Find the area y = x 2 and the x-axis [0,1] using 4 partitions and the trapezoids. Trapezoids 0 1/4 1/2 3/4 1 Why were areas found using RM only estimates? How could we draw lines to improve our estimates? What shape do you get? 0 1/4 1/2 3/4 1 Slide 29 / 163 *NOTE: Trapezoid pproximation = LRM + RRM 2 Slide 30 / 163 10 The area between and the x-axis [1,3] is approximated with 4 partitions and trapezoids. What is the height of each trapezoid? 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 P alc exam.
Slide 31 / 163 Slide 32 / 163 11 The area between and the x-axis [1,3] is approximated with 4 partitions and trapezoids. What is the area of the 4 th trapezoid? 12 The area between y = and the x-axis [1,3] is approximated with 4 partitions and trapezoids. What is the approximate area? Slide 33 / 163 Slide 34 / 163 13 What is the approximate area using the trapezoids that are drawn? 14 What is the approximate fuel consumed using the trapezoids rule for this hour flight? Time (minutes) Rate of onsumption (gal/min) 0 0 10 20 25 30 40 40 60 45 Slide 35 / 163 In the last 2 responder questions, the partitioned intervals weren't uniform. The P will use both. So don't assume. Slide 36 / 163 So far we have been summing areas using #. Gottfried Leibniz, a German mathematician, came up with a symbol you're going to see a lot of: #. It is actually the German S instead of the Greek. It still means summation. s a point of interest, we use the German notation in calculus because Leibniz was the first to publish. Sir Isaac Newton is now given the credit for unifying calculus because his notes predate Leibniz's.
Slide 37 / 163 Slide 38 / 163 ccumulation Function ccumulation Function V (m/s) nother way we can calculate area under a function is to use geometry. t (sec) What's happening during t=0 to t=3? What is the area of t=0 to t=3? What does the area mean? What is the acceleration at t=3? Return to Table of ontents What is happening during t=3 to t=6? What is the area of t=3 to t=6? What does this area mean? Where is the object at t=6 in relation to where it was at t=0? Slide 39 / 163 Slide 40 / 163 ccumulation Function The symbol notation for the area from zero to 3: "The area from t=0 to t=3 is the integral from 0 to 3 of the velocity function with respect to t." In general: Slide 41 / 163 Slide 42 / 163 ccumulation Function When solving an accumulation function: (direction)(relation to x-axis)(area) ccumulation Function 15 semicircle
Slide 43 / 163 Slide 44 / 163 ccumulation Function ccumulation Function 16 (round to two decimal places) 17 (round to two decimal places) semicircle semicircle Slide 45 / 163 Slide 46 / 163 ccumulation Function 18 semicircle Slide 47 / 163 Slide 48 / 163 rea under the curve of f(x) from a to b is & efinite Integrals We have been using geometry to find. The antiderivative of f(x) can also be used. Return to Table of ontents
Slide 49 / 163 Slide 50 / 163 Properties of efinite Integrals 20 Slide 51 / 163 Slide 52 / 163 21 22 Slide 53 / 163 Slide 54 / 163 23 ntiderivative Rules Why +?
Slide 55 / 163 Slide 56 / 163 Slide 57 / 163 Slide 58 / 163 24 25 Slide 59 / 163 Slide 60 / 163 26 27
Slide 61 / 163 Slide 62 / 163 28 29 Slide 63 / 163 Slide 64 / 163 30 Slide 65 / 163 Slide 66 / 163 The graphing calculator also has a built-in integration function. MTH -> 9:fnInt( depending on the version of the operating system: fnint( or For example, integrate x 2-3 from 1 to 4 with respect to x. fnint(x 2-3,x,1,4) Fundamental Theorem of alculus Return to Table of ontents
Slide 67 / 163 Slide 68 / 163 Fundamental Theorem of alculus Fundamental Theorem of alculus (F.T..) Part 2 If F(x) is continuous at every point of [a,b] then has a derivative at every point on [a,b],and Slide 69 / 163 Slide 70 / 163 Fundamental Theorem of alculus Fundamental Theorem of alculus Example: 31 It looks easy, but be aware. When the derivative of the bounds in anything other that 1, need to multiply f(x) by the derivative. Slide 71 / 163 Slide 72 / 163 Fundamental Theorem of alculus Fundamental Theorem of alculus 32 33
Slide 73 / 163 Slide 74 / 163 Fundamental Theorem of alculus 34 HINT Return to Table of ontents Slide 75 / 163 Slide 76 / 163 Just like with differentiation, there are many integrals that are more complicated to evaluate. In situations like these, we use the to turn a difficult integral into a much simpler one. Slide 77 / 163 Slide 78 / 163 Ex: Ex: Let Let
Slide 79 / 163 Slide 80 / 163 35 What is the value of u? 36 What will the integral be after the substitution is made? Slide 81 / 163 Slide 82 / 163 37 Evaluate the integral The If you have a definite integral, you have two options for plugging in the bounds to get the final answer. 1. Plug a and b into the integrated function FTER you have re-substituted the x's back into the function 2. Plug a and b into to create two new bounds, and plug these into the integrated function Slide 83 / 163 Slide 84 / 163 Ex: 38 What is the value of u? Let
Slide 85 / 163 39 What is the new upper bound? Slide 86 / 163 40 What is the new lower bound? Slide 87 / 163 Slide 88 / 163 41 What will the integral be after the substitution is made? 42 What is the value of the integral? Slide 89 / 163 Slide 90 / 163 43 What is the value of u? 44 What is the new upper bound?
Slide 91 / 163 Slide 92 / 163 45 What is the new lower bound? 46 What will the integral be after the substitution is made? Slide 93 / 163 Slide 94 / 163 47 What is the value of the integral? rea etween urves Return to Table of ontents Slide 95 / 163 Slide 96 / 163
Slide 97 / 163 Slide 98 / 163 rea etween urves rea etween urves 48 When finding the area between and, what is the left bounds of x? 49 When finding the area between and, what is the right bounds of x? Slide 99 / 163 Slide 100 / 163 rea etween urves rea etween urves 50 When finding the area between and, what is integral used? 51 What is the area between and? Slide 101 / 163 Slide 102 / 163
Slide 103 / 163 Slide 104 / 163 Slide 105 / 163 Example: Find the area between the curves in the first quadrant. Notice that the lower function isn't the same for the entire region. We could find the area in terms of y or divide the region into 2 separate integrals and add their areas. rea etween urves 55 Slide 106 / 163 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what would be the area to the left of the y-axis? Slide 107 / 163 Slide 108 / 163 rea etween urves 56 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what is the left bounds of h(x) - f(x)? rea etween urves 57 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what is the right bound of h(x) - f(x)?
rea etween urves 58 Slide 109 / 163 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what would be the area to the right of the y-axis? rea etween urves Slide 110 / 163 59 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what is the right bound of h(x) - g(x)? Slide 111 / 163 Slide 112 / 163 rea etween urves 60 When finding the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1, what is the left bound of h(x) - g(x)? rea etween urves 61 Find the area between f(x) = x, g(x) = -1/2x, and h(x) =1/2x - 1. Slide 113 / 163 Slide 114 / 163 nother way to make a 3- object is to take a region and rotate it about an axis. Volume: isk Method When this rectangle is rotated a cylinder is formed. We could use geometry, but can we use calculus? Return to Table of ontents From the section on known cross sections: nd since the cross sections are circles:
Slide 115 / 163 Slide 116 / 163 Rotate about x-axis from x=1 to x=4. When rotating about a horizontal axis: When rotating about a vertical axis: Slide 117 / 163 Slide 118 / 163 Rotate about y=2 from x=1 to x=4. Since y=2 is a horizontal axis of rotation, integral is in terms of x. Rotate about y= -2 from x=1 to x=4. Since y=-2 is a horizontal axis of rotation, integral is in terms of x. y -2 r=2-y Why? From the x-axis to the curve is y and to axis of rotation is 2, we want upper minus lower. r=y- -2=y+2 Why? upper minus lower Slide 119 / 163 Slide 120 / 163 62 Rotate y=2x 2 about the x-axis over [0,5]. What is the lower bound? 63 Rotate y=2x 2 about the x-axis over [0,5]. What is the upper bound?
Slide 121 / 163 Slide 122 / 163 64 Rotate y=2x 2 about the x-axis over [0,5]. What is integral? 65 Rotate y=2x 2 about the x-axis over [0,5]. What is the volume? Slide 123 / 163 Slide 124 / 163 66 Rotate y=2x 2 about the line x=4 over [0,5]. What is the lower bound? 67 Rotate y=2x 2 about the line x=4 over [0,5]. What is the upper bound? Slide 125 / 163 Slide 126 / 163 68 Rotate y=2x 2 about the line x=4 over [0,5]. What is the radius? 69 Rotate y=2x 2 about the line x=4 over [0,5]. What is integral?
Slide 127 / 163 70 Rotate y=2x 2 about the line x=4 over [0,5]. What is the volume? Slide 128 / 163 Rotate for x=0 to x=2 about the y-axis. Since this is a vertical axis,the problem should be rewritten in terms of y: Rotate for y=0 to y=16 about the y-axis. 16 Slide 129 / 163 Slide 130 / 163 Rotate for x=0 to x=2 about the x=2. Since this is a vertical axis, the problem should be rewritten in terms of y: 71 Rotate about the y-axis over. What is the lower bound? 16 Rotate for y=0 to y=16 about the x=2. Slide 131 / 163 Slide 132 / 163 72 Rotate about the y-axis over. What is the upper bound? 73 Rotate about the y-axis over. What is the radius?
Slide 133 / 163 Slide 134 / 163 74 Rotate about the y-axis over. What is integral? 75 Rotate about the y-axis over. What is the volume? Slide 135 / 163 Slide 136 / 163 Slide 137 / 163 Slide 138 / 163
Slide 139 / 163 Slide 140 / 163 Volume: Washer Method Return to Table of ontents Slide 141 / 163 Slide 142 / 163 For the washer method, there is a gap between the region being rotated and the axis. When rotating about a horizontal axis: Rotating this rectangle we get a tube, or a cylinder with a smaller cylinder taken away. Where R is the greater distance from the axis, not necessarily the upper function. When rotating about a vertical axis: Our cross section would be: R r *aution: # can be factored out but not the square. Slide 143 / 163 Slide 144 / 163 Find the volume when f(x) and g(x) are rotated about the x-axis [0,2]. Rotate the region bound by y=x 2, x=2, and y=0 about the y-axis. Since a vertical axis of rotation, integration is done in terms of y. (2,4) Hint
Slide 145 / 163 Slide 146 / 163 Rotate the region bound by x-axis, y=x 2, x=1, and x=2 about y= -1. Since y= -1 is horizontal integration is done in terms of x. 81 Rotate the region between and about the x-axis over [0,1]. What is the lower bound? r=1 R=x 2 +1 Slide 147 / 163 Slide 148 / 163 82 Rotate the region between and about the x-axis over [0,1]. What is the upper bound? 83 Rotate the region between and about the x-axis over [0,1]. What is integral? Slide 149 / 163 Slide 150 / 163 84 Rotate the region between and about the x-axis over [0,1]. What is the volume? 85 Rotate the region between and about the y-axis over [0,1]. What is integral?
Slide 151 / 163 Slide 152 / 163 86 Rotate the region between and about the y-axis over [0,1]. What is the volume? 87 Rotate the region between and about the y=1 over [0,1]. What is R? Slide 153 / 163 Slide 154 / 163 88 Rotate the region between and about the y=1 over [0,1]. What is integral? 89 Rotate the region between and about the y=1 over [0,1]. What is the volume? Slide 155 / 163 Slide 156 / 163 90 Rotate the region between and about the x=-1 over [0,1]. What is integral? 91 Rotate the region between and about the x=-1 over [0,1]. What is the volume?
Slide 157 / 163 Slide 158 / 163 Volume: Shell Method Volume: Shell Method Volume: Shell Method When rotating about a horizontal axis: Return to Table of ontents When rotating about a vertical axis: Slide 159 / 163 Slide 160 / 163 Volume: Shell Method Find the volume of the solid obtained by rotating the area between the graph of and about the y-axis. Slide 161 / 163 Slide 162 / 163 Volume: Shell Method 92 Find the volume of the solid generated by rotating the area under the graph of from about the y-axis.
Slide 163 / 163 Volume: Shell Method 93 Find the volume of the solid generated by rotating the area between the graphs of and about the y-axis.