How to solve an AP Calculus Problem…

Jon Madara, Mark Palli, Eric Rakoczy

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

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.

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.

dxxfxgb

a

])()([

17821805.

0

)sin(4

1)4( dxxx

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!

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.

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.

dxx x )]4()sin(4

1[

1

17821805.

dxxgxfb

a )]()([

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!

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.

b

a

xgxf 22 )(1)(1

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.

dxx x ])4(1]sin4

11[

1

17821805.

22

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!