6.1 Area Between Curves. Example 1: Calculate the area of the region between the parabola y = 1 x 2 and the line y = 1 x

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1 AP Calculus 6.1 Area Between Curves Name: Goal: Calculate the Area between curves Keys to Success: Top Curve Bottom Curve (integrate w/respect to x or dx) Right Curve Left Curve (integrate w/respect to y or dy) Change integrals when curves change positioning Example 1: Calculate the area of the region between the parabola y = 1 x 2 and the line y = 1 x Example 2: Calculate the area of the region between y = sin x, y = cos x, x = 0, x = π/2. Example 3: Calculate the area of the region between y = x 1 and y 2 = 2x + 6

2 Directions: Show all work including graph. Give exact answers, but check with a calculator. 1. Find the area between the x-axis and the curve y 2 2 x from 0 x to x Find the area of the region between the curve x = y 2 6 and x = y.

3 AP Calculus AB 6.2 Volumes of Solids of Revolution Name: Objective: Calculate the volume of a 3 dimensional shape created by revolving a 2 dimensional graph around an axis of revolution. Method: By calculating the volume of infinitely thin circular cross-sections, called disks and washers, we can calculate the volume of a 3-dimensional shape by adding the volume of each disk or washer. Visualization: The line y = 3x revolved around the x-axis, creates a cone. The curve y = x revolved about the x-axis. y = x Websites to help with visualization: Exercise 1: Calculate the volume of the solid formed by revolving the line y = 3x about the x-axis from 0 x 1. (a) Using the volume formula for a cone (b) Using calculus and the disk-method

4 Disk-Method Horizontal axis of rotation b π(f(x)) 2 dx a f(x) is the radius of each disk and dx is the infinitesimal width Vertical axis of rotation b π(f(y)) 2 dy a f(y) is the radius of each disk and dy is the infinitesimal width Exercise 2: Using the disk-method, calculate the volume of the solid formed by revolving y = x about the x-axis from 0 x 4 Exercise 3: Calculate the volume of the solid formed by revolving y = (x 1) 2 about the x-axis from 2 x 3. Exercise 4: Calculate the volume of the solid obtained by rotating the region bounded by y = x 3, y = 8, and x = 0 about the y-axis.

5 6.2 Notes (continued) Washer Method If the curve is not completely against the axis of revolution, then there will be a section of the solid that doesn t have any volume. Volume of a washer: V = πr 2 h πr 2 h Exercise 5: (a) Find the volume of the solid obtained by revolving the region bounded by y = 3x, y = x 2, x 0 about the x axis. (b) Find the volume of the solid obtained by revolving the region bounded by y = 3x, y = x 2, x 0 about the line y = 2.

6 (c) Find the volume of the solid obtained by revolving the region bounded by y = 3x, y = x 2, x 0 about the line y = 9.

7 Volumes of Solids of Known Cross Sections Calculus Concepts Name Date Let R be the region bounded by the graphs of x 2 = y and x = 9. Find the volume of the solid that has R as its base if every cross section by a plane perpendicular to the x -axis has the given shape. 1. A square 2. A rectangle of height 2 3. A semicircle 4. A quarter circle Calculus, 6 th edition Swokowski

8 5. An equilateral triangle 6. A triangle with height equal to ¼ the length of the base. 7. A trapezoid with lower base in the xy -plane, upper base equal to ½ the length of the lower base, and height equal to ¼ the length of the lower base. Answers: π π

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10 AP Calculus 6.5 Mean Value Theorem for Integrals Name: We already learned the MVT for derivatives. It said, For a continuous function, the average rate of change must equal the instantaneous rate of change at least once. Exercise 1: Find the numbers in the interval that satisfy the mean value theorem for f(x) = 3x 2 + 2x + 5, [ 1,4] The MVT for integrals is similar. It says, For a continuous function, the area under a curve can also be found by finding the area under the average value line. 3 Exercise 2: Find the average value of the function f(x) = x, [1,8] Basic Meaning: MVT for Derivatives Your average speed in MPH for a trip means that for at least one moment you were traveling your average speed. MVT for Integrals If you drove your average speed for the whole trip, then it would take the same amount of time to drive that distance on cruise control as opposed to changing speeds. Exercise 3: Suppose it takes 3 minutes to drive 2.6 miles with varying speeds. What speed could you set your cruise control on so that you also drive 2.6 miles in exactly 3 minutes? Remember: MVT for derivatives = Big brother speeding ticket MVT for integrals is the tortoise and hare (except they tie)

11 Mean value theorem for Integrals You are driving the 37.2 miles from school to Chicago and it takes you 1 hr and 23 minutes. Your speed is obviously changing throughout the trip due to traffic, stoplights, and the occasional stranded car on the side of I-55. Can you find your average speed in miles per hour? If you set your cruise control to the average speed, how long would it take to make the trip? Now, imagine you are driving to Chicago at 3 in the morning and there is no traffic, all green lights, and no bad drivers. Find the speed you would have to set your cruise control for it to take your exactly 25 minutes and 45 seconds to make the trip. Exercise 4: Calculate the average value of f(x) = x 1 x 4 Exercise 5: Calculate the average value of f(x) = sec 2 x 0 x π 4 Exercise 6: Let f(x) = 4x 2, [2,5]. Find c such that fave = f(c).