Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the curves about the given axis.y = 8ex, y = 8e−x, x = 1; about the y-axis

Answer 1

See attachment for one such shell. The volume is given by the sum of infinitely many thin shells like this, each with radius $x$ and height determined by the vertical distance between the upper blue curve and the lower orange curve for any given $x$, i.e. $8e^x-8e^{-x}$.

The volume is then

$2\pi\displaystyle\int_0^1x\left(8e^x-8e^{-x}\right)\,\mathrm dx=16\pi\int xe^x(e^{2x}-1)\,\mathrm dx=\dfrac{32\pi}9(e^3-4)\approx179.677$