Finding cross sectional volume

Question

Can someone please help me solve this problem?

Q: A volume is above the area bounded by the curves $y = x^2$ and $y = 1$ in the $xy$-plane. Each $x$ cross section is a rectangle with base touching the ends of the curves and height twice the base. If $0 < x < 1$, what is the volume?

This is what i have so far, the $A(x)$ should be equal to $= 2(1-x^2)^2$ but I am not even sure if that is right.

Is that the correct formula? If so, how do I integrate that?

Answer 1

You did it right. See the following plot:

Now, we have $$V=\int_0^1(1-x^2)\times (2\times(1-x^2))dx=2\int_0^1(1-x^2)^2dx\\=2\int_0^1(1-2x^2+x^4)dx=2[x-\frac{2}{3}x^3+\frac{x^5}5]_0^1=2(1-2/3+1/5)-2(0-0+0)$$