Question

Question #ae4f3

Answer 1

The volume is `53.617`.

Explanation:

There is only one curve being rotated, so we can use the disc method. The disc method says that for each value of `x`, the vertical cross section is a circle with an area of `pi*f(x)^2`, since `f(x)` is the radius. Therefore:

`V = int_a^b pi*f(x)^2 dx = pi* int_a^b f(x)^2 dx`

First, we need to find our bounds. Since we are given no other bounds, the bounds must be the zeroes of `y = 4-4x^2`. So, we set `y` equal to zero and solve for our two `x` values.

`0 = 4-4x^2`
`4x^2 = 4`
`x^2 = 1`
`x = +-1`

So, our bounds are `-1` and `1`.

Now, all we have left to do is use the disc method formula to find the volume.

`V = pi* int_a^b f(x)^2 dx`

`= pi * int_-1^1 (4-4x^2)^2 dx`

`= pi * int_-1^1 4^2 * (1-x^2)^2 dx`

`= 16pi * int_-1^1 (x^4 - 2x^2 + 1) dx`

`= 16pi * (x^5/5-2x^3/3+x)|_-1^1`

`= 16pi * ((1/5 - 2/3+1) - (-1/5 + 2/3-1))`

`= 16pi * (2/5-4/3+2)`

`=(256pi)/15`

`= 53.617`

Final Answer