Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis. y = 13 e**(-x**2) text(, ) y = 0 text(, ) x = 0 text(, ) x = 1 V = Sketch the region and a typical shell. (Do this on paper. Your instructor may ask you to turn in this sketch.)

Answer 1

Answer:

As per the given statement:

The region bounded by the given curves about the y-axis, $y = 13e^{-x^2}$, y=0, x = 0 and x = 1

Using cylindrical shell method:

The volume of solid(V) is obtained by rotating about y-axis and the region under the curve y = f(x) from a to b is;

$V = \int_{a}^{b} 2\pi x f(x) dx$   where $0\leq a$

where x is the radius of the cylinder

f(x) is the height of the cylinder.

From the given figure:

radius = x

height(h) =f(x) =y=$13e^{-x^2}$

a = 0 and b = 1

So, the volume V generated by rotating the given region:

$V =2 \pi \int_{0}^{1} x ( 13e^{-x^2}) dx\\\\V=2\pi\left [ -\frac{13}{2}e^{-x^2} \right ]_{0}^{1}\\\\V=2\pi\left (-\frac{13}{2e}-\left(-\frac{13}{2}\right) \right )\\\\V=-\frac{13\pi }{e}+13\pi$

therefore, the volume of V generated by rotating the given region is $V=-\frac{13\pi }{e}+13\pi$