Area and Volume Formulas — Question 8

Question 8 - Solution

We derive the volume formula using cross-sectional areas and integration.

Place the cylinder so that its base lies in the $xy$-plane and its axis is along the $x$-axis. Then the cylinder extends from $x=0$ to $x=h$.

At any position $x$ with $0\le x\le h$, a cross-section perpendicular to the $x$-axis is a circle of radius $R$.

The area of this circular cross-section is constant and equal to $$A(x)=\pi R^2.$$

The volume of the cylinder can be approximated by slicing it into thin slabs of thickness $dx$. Each slab has volume approximately $$dV=A(x)\,dx=\pi R^2\,dx.$$

Adding the volumes of all such slabs gives $$V=\int_0^h \pi R^2\,dx.$$

Since $\pi R^2$ is constant, evaluate the integral: $$V=\pi R^2\int_0^h dx =\pi R^2[x]_0^h =\pi R^2 h.$$

$$\boxed{V=\pi R^2 h}$$