# How do you find the volume of a pyramid using integrals?

Oct 2, 2014

Let us find the volume of a pyramid of height $h$ with a $b \setminus \times b$ square base.

If $y$ is the vertical distance from the top of the pyramid, then the square cross-sectional area $A \left(y\right)$ can be expressed as

$A \left(y\right) = {\left(\frac{b}{h} y\right)}^{2} = {b}^{2} / {h}^{2} {y}^{2}$.

So, the volume $V$ can be found by the integral

V=int_0^hA(y) dy=b^2/h^2int_0^hy^2 dy=b^2/h^2[y^3/3]_0^h =1/3b^2h.

I hope that this was helpful.