Let be the region in the plane bounded by the curves ,

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

Problem

Let

S

be the region in the plane bounded by the curves

y = x^2

y = 2x - x^2

, and the

y

–axis. The entire region is revolved about the

y

–axis. (a) Sketch a clear labeled diagram of the region

S

, indicating the points of intersection of

y = x^2

and

y = 2x - x^2

. (b) Express the

y

–coordinates of the intersection points found in part (a) in exact form. (c) Write the integral expression(s) that represent the volume of the solid obtained when region

S

is revolved about the

y

–axis. Your integrals should use horizontal slices and may require more than one integral. (d) Evaluate the integral expression(s) from 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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