How do you use the method of cylindrical shells to find the volume generated by rotating the region bounded by $y = e^{- x^{2}}$ $y=e^(−x^2)$ , y=0, x=0, and x=1 about the y axis?

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Answer 1 · 2015-10-12T20:14:05

Draw a sketch, then integrate using the shell method with respect to x.

Here is sketch of the problem:

enter image source here

Using the formula for the Shell Method:

$2 π \int_{0}^{1} (x) (e^{- x^{2}} - 0) d x$ $2piint_0^1(x)(e^(-x^2)-0)dx$

You can use substitution { $u = x^{2}$ $u=x^2$ } to solve.

$= π (1 - e^{- 1})$ $=pi(1-e^-1)$

Answer $\approx 1.986$ $~~1.986$

Hope that helps

How do you use the method of cylindrical shells to find the volume generated by rotating the region bounded by y=e−x2y=e^(−x^2), y=0, x=0, and x=1 about the y axis?