Question #ae4f3

1 Answer
Mar 26, 2017

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