For the curve describes the right half of a circle of radius centered at the origin

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

Problem

For

0\le y\le 3

the curve

x=\sqrt{9-y^{2}}

describes the right half of a circle of radius

3

centered at the origin. Region

S

is the set of points between this curve and the

y

-axis. Region

T

is the triangular region bounded by the

y

-axis, the line

y=3

, and the line

x=3y/5

. Each region is revolved about the

y

-axis, creating solids with volumes

V_S

and

V_T

, respectively. (a) Set up, but do not evaluate, a definite integral in terms of

y

whose value is

V_S

. (b) Set up, but do not evaluate, a definite integral in terms of

y

whose value is

V_T

. (c) Without evaluating either integral, determine whether

V_S

is less than, equal to, or greater than

V_T

. Provide a reasoned comparison based on the radii of corresponding cross-sections. (d) Describe one change to region

T

that would make the volumes

V_S

and

V_T

equal, and justify why your modification achieves equality.