A region S in the first quadrant is enclosed by the graphs of the curves • • • the…

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

Problem

A region S in the first quadrant is enclosed by the graphs of the curves • y=xy = \sqrt{x} • y=x4y = \dfrac{x}{4} • the vertical line x=4x = 4 The region S is revolved about the horizontal line y=−1y = -1, generating a solid of revolution. (a) Determine the xx–coordinate of the point where the curves y=xy = \sqrt{x} and y=x4y = \dfrac{x}{4} intersect on the interval 0≤x≤40 \le x \le 4. (b) Write, but do not evaluate, a definite integral expression (using one or more integrals) that gives the exact volume of the solid obtained by revolving SS about the line y=−1y = -1. Be sure to indicate the limits of integration for each integral and clearly identify which integrand corresponds to which portion of the region.

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