Volumes with Known Cross-Sections

$V = \int_a^b A(x)\,dx$

where $A(x)$ is the area of the cross-section at position $x$.

If the base extends from $f(x)$ to $g(x)$, the side length is $s = f(x) - g(x)$.

Cross-Section	Area
Square	$s^2$
Semicircle	$\frac{\pi}{8}s^2$
Equilateral triangle	$\frac{\sqrt{3}}{4}s^2$
Isosceles right triangle	$\frac{1}{2}s^2$

Base: region between $y = \sqrt{x}$ and $y = 0$ from $x = 0$ to $x = 4$. Cross-sections perpendicular to x-axis are squares.

Side = $\sqrt{x}$. $A(x) = (\sqrt{x})^2 = x$.

$V = \int_0^4 x\,dx = 8$.