Question

How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by `y = 1/x^4`, y = 0, x = 1, x = 4 revolved about the x=-4?

Answer 1

`(57pi)/16`

Explanation:

the formula for the shell method is `int_a^b2pirhdx`

`a` and `b` are the x-bounds, which are x=1 and x=4, so `a=1` and `b=4`.

`r` is the distance from a certain x-value in the interval `[1,4]` and the axis of rotation, which is x=-4. `r=x-(-4)=x+4`

`h` is the height of the cylinder at a certain x-value in the interval `[1,4]`, which is `1/x^4-0=1/x^4` (because `1/x^4` is always greater than `0` and h must be positive).

plugging it all in: volume `=int_1^4(2pi(x+4)(1/x^4))dx`
you should get: `(57pi)/16`