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How to solve an AP Calculus Problem…

How to solve an AP Calculus Problem…. Jon Madara, Mark Palli, Eric Rakoczy.

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How to solve an AP Calculus Problem…

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  1. How to solve an AP Calculus Problem… Jon Madara, Mark Palli, Eric Rakoczy

  2. 1. Let f and g be the functions given by f(x)= ¼ +sin(πx) and g(x)= 4-x. Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Find the area of S. (c) Find the volume of the solid generated when S is revolved about the horizontal line y = -1

  3. Area of R • First, we must find the point of intersection between f(x) and g(x). To do this, enter the two equations into a calculator and graph them. Then use the intersect function to calculate the point of intersection.

  4. Finding Area of R • To find the area of R, take the integral of the difference between the upper and lower functions. • Substitute in the upper function for g(x), the lower function for f(x), and the end points of the interval.

  5. Finding Area of R • Enter the resulting equation into your calculator. • Find the integral from 0 to 0.17821805 • And the answer is: • 0.065!

  6. Finding Area of S • First, we must find the second point of intersection. (We have the first point from the last part) To do this, enter the two equations into a calculator and graph them. Then use the intersect function to calculate the second point of intersection.

  7. Finding Area of S • To find the area of S, take the integral of the difference between the upper and lower functions. • We substituted in the upper function for f(x), the lower function for g(x), and the new end points of the interval.

  8. Finding Area of S • Enter the equation inside the integral into your calculator. • Use the calculator to find the integral from 0.17821805 to 1. • And the answer is: • 0.4104!

  9. Finding Volume of Solid • We can use the disk and washer method to find the volume of the solid formed when S is rotated around the horizontal line y = -1. • The endpoints of the interval for section S are the same as from the previous question.

  10. Finding Volume of Solid • For this question, you need to add 1 to f(x) and g(x), since the area is rotated around the line y = -1.

  11. Finding Volume of Solid • Enter the equation inside the integral into your calculator. • Use the calculator to find the integral from 0.17821805 to 1. • Multiply the integral by pi. • And the answer is: • 4.558!

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