Consider the region that lies inside the circle and to the right of the vertical line

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Consider the region that lies inside the circle

x^{2}+y^{2}=25

and to the right of the vertical line

x=3

. This region serves as the base of a solid. For each value of

y

in the base, the cross section perpendicular to the

y

-axis is a rectangle whose height is four times its base. (a) Find the

y

-coordinates where the circle

x^{2}+y^{2}=25

intersects the line

x=3

. (b) Express the length of the base of a typical rectangular cross section as a function of

y

. (c) Write an integral that represents the volume of the solid. (d) Evaluate the integral to obtain the exact volume.