A region in the first quadrant is bounded on the left by

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

Problem

A region

S

in the first quadrant is bounded on the left by

x=y+2

, on the right by

x=10-y^{2}

, below by the

y

–axis intersection at

y=0

, and above by

y=3

. When

S

is revolved about the

y

–axis, a solid is formed. (a) Determine all values of

y

[0,3]

where the curves

x=y+2

and

x=10-y^{2}

intersect. (b) The relative position of the two curves changes within

[0,3]

. Identify the sub–intervals of

y

where each curve is farther from the

y

–axis, supporting your answer analytically. (c) Write a piecewise–defined definite integral that represents the volume of the solid obtained by revolving

S

about the

y

–axis. (d) Evaluate the integral(s) found in part (c) to obtain the exact volume of the solid.

Related theory

Area Between Curves
Area Applications
Volume Applications
Applications of Integration

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