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katrin [286]
3 years ago
14

The base of a solid in the region bounded by the graphs of y=e^-xy=0, and x=0, and x=1. cross sections of the solid perpendicula

r to the x-axis are semicircles. what is the volume, in cubic units, of the solid?
Mathematics
1 answer:
svlad2 [7]3 years ago
6 0

The diameter of each semicircular cross-section is given by the vertical distance between the curves y=e^{-x} and y=0, which is just e^{-x}.

Given a semicircle with diameter d, its area is \dfrac{\pi d^2}8, so the area of each semicircular section is \dfrac{\pi e^{-2x}}8.

The volume of the solid is obtained by integrating from 0 to 1:

\displaystyle\frac\pi8\int_{x=0}^{x=1}e^{-2x}\,\mathrm dx=\frac{(e^2-1)\pi}{16e^2}

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