Volume of Solid W/ Base Region...

Question

How would I go about the following?

Find the volume of the solid $W$ whose base is the region enclosed by $y=x^2$ and $y=1$, and the cross-sections perpendicular to the y-axis are squares.

Answer 1

At any position $x$ you have a square of side $1-x^2$. Therefore, you can sum the areas of the squares along $x$. The figure below may help. Thus,

$$V=\int_{-1}^1 (1-x^2)^2~dx=\frac{16}{15}$$

Answer 2

For each $y$ fixed $x$ varies from $-\sqrt y$ to $\sqrt y$, so the integral is: $$ \int_0^1\int_{-\sqrt y}^{\sqrt y}\int_0^{2\sqrt y}\,dz\,dx\,dy = \cdots $$