Consider the solid obtained by revolving the region in the xy–plane that lies between…

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

Problem

Consider the solid obtained by revolving the region in the xy–plane that lies between the graphs of y=e−xy = e^{-x} and y=e−2xy = e^{-2x} over the interval 0≤x≤ln⁡20 \le x \le \ln 2 about the x–axis. (a) Prove that e−x≥e−2xe^{-x} \ge e^{-2x} for all xx in [0,ln⁡2][0,\ln 2], and hence identify which function determines the outer radius of a typical cross-section after the rotation. (b) Write an exact definite integral expression for the volume of the solid. (c) Evaluate the integral from part (b) exactly. Simplify your final answer and include appropriate units.

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