Let be the closed region in the –plane bounded by , ,

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

Problem

Let

S

be the closed region in the

xy

–plane bounded by

x = y^2

x = 2y

y = 0

, and

y = 4

. The region is revolved about the vertical line

x = -1

to create a solid. (a) Find the

y

-coordinates of all intersection points of the curves

x = y^2

and

x = 2y

inside the interval

0 \leq y \leq 4

. (b) On the interval

0 \leq y \leq 4

, determine the subintervals on which the curve

x = y^2

is farther from the axis

x = -1

than

x = 2y

. Clearly state the outer and inner boundaries on each subinterval. (c) Write, but do not yet evaluate, the sum of definite integrals with respect to

y

that represents the volume of the solid. (d) Evaluate the integrals from part (c) to obtain the exact volume.