Computing Volume Using Integration

Question

The area of the base of the shape is enclosed by the functions $f(x)=\sqrt{x}$ and $g(x)=x$, and whose cross-sections are squares.

I'm having trouble imagining what this looks like let alone computing its volume, any help would be much appreciated.

Answer 1

The curves $f$ and $g$ intersect at $x=0$ and $x=1$. A cross-section of the solid taken perpendicular to the $x$ axis at some point $x$ will have side length $s$ where $s=x-\sqrt{x}$. Hence the area of this cross section will be $A(x)=s^2=(x-\sqrt{x})^2$. Thus the volume is $$ \int_{0}^1 A(x)\, dx=\int_{0}^1 (x-\sqrt{x})^2\, dx. $$