A region in the –plane is bounded on the right by the curve

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

Problem

A region in the

xy

–plane is bounded on the right by the curve

x=\sqrt{y+1}

, on the left by the

y

–axis, and vertically between

y=-1

and

y=3

. The region is revolved about the

y

–axis to generate a solid of revolution. (a) Describe, in terms of

y

, the distance from a typical point on the curve to the axis of rotation. (b) Explain why the volume

V

of the solid can be expressed in the form

V=\pi\displaystyle \int_{-1}^{3}\!(\,\cdots\,)^{2}\,dy

. Identify the integrand explicitly. (c) Interpret geometrically what the quantity

(\,\cdots\,)^{2}

inside the integral represents and justify the presence of the square.