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

3. 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

4. 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.

5. 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.

6. 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!

7. 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.

8. 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.

9. 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!

10. 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.

11. 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.

12. 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!