Question

Calc II Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y axis.
Y = e^(-x^2)
Y = 0
X = 0
X = 1

Correct answer is pi (1 - (1/e))
I'm just not sure how to get to that answer

Answer 1

`\displaystyle \pi\biggr(1-(1)/(e)\biggr)`

Explanation:

Shell Method (Vertical Axis)

`\displaystyle V=2\pi\int^b_ar(x)h(x)\,dx`

Radius:
`r(x)=x`

Height:
`h(x)=e^(-x^2)`

Bounds:
`[a,b]=[0,1]`

Set up and evaluate integral

`\displaystyle V=2\pi\int^1_0xe^(-x^2)\,dx`

• Let
  `u=-x^2` and
  `du=-2x\,dx` so that
  `-(1)/(2)\,du=x\,dx`

• Bounds become
  `u=-0^2=0` and
  `u=-1^2=-1`

`\displaystyle V= -(1)/(2)\cdot2\pi\int^(-1)_0e^u\,du\\\\V= -\pi\int^(-1)_0e^u\,du\\\\V=\pi\int^0_(-1)e^u\,du\\\\V=\pi e^u\biggr|^0_(-1)\\\\V=\pi e^0-\pi e^(-1)\\\\V=\pi-(\pi)/(e)\\\\V=\pi\biggr(1-(1)/(e)\biggr)`