Consider the region R in the first quadrant that is bounded by the curves and

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

Problem

Consider the region R in the first quadrant that is bounded by the curves

y=x^{2}

and

y=2x

. The region is revolved about the

y

-axis to produce a solid of revolution. Without carrying out any numerical integration, analyze how to compute the volume of the solid. (a) Sketch or describe a sketch of region R, clearly indicating the points of intersection of the two bounding curves. (b) Rewrite each bounding curve as

x

expressed in terms of

y

, and determine the

y

-interval over which the region exists. (c) For each

y

in the interval found in part (b), identify which curve determines the outer radius and which determines the inner radius of the solid. Provide a brief explanation that refers to distances from the axis of revolution. (d) Write—but do not evaluate—a definite integral expression that represents the exact volume of the solid generated by revolving R about the

y

-axis.

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