Volume of a solid with square cross-sections

Text

A solid has as its base the region under $y=\sqrt{x}$ for $0\le x\le4$; cross-sections perpendicular to the $x$-axis are squares with side equal to the ordinate $y=\sqrt{x}$. Compute the volume.

Solution

Cross-section area: $A(x)=\big(\sqrt{x}\big)^{2}=x$. $V=\displaystyle\int_0^4 x\,dx=\Big[\tfrac{x^{2}}{2}\Big]_0^4=8$. $\boxed{V=8\ \text{u}^3}$