# How do you find the volume of the solid with base region bounded by the curve y=1-x^2 and the x-axis if cross sections perpendicular to the y-axis are squares?

Sep 22, 2014

The volume is 2.

Let us look at some details.

By rewriting,

$y = 1 - {x}^{2} \Leftrightarrow x = \pm \sqrt{1 - y}$

Since the length of the side of each cross sectional square is $2 \sqrt{1 - y}$, the cross sectional area $A \left(y\right)$ can be given by

$A \left(y\right) = {\left(2 \sqrt{1 - y}\right)}^{2} = 4 \left(1 - y\right)$

Since the base spans from $y = 0$ to $y = 1$, the volume $V$ can be found by

$V = 4 {\int}_{0}^{1} \left(1 - y\right) \mathrm{dy} = 4 {\left[y - {y}^{2} / 2\right]}_{0}^{1} = 4 \left(1 - \frac{1}{2}\right) = 2$