# Volumes by Cross Sections

Volumes by Cross Sections can be setting a unit volume, $\delta V$ of a cross section. Then establish an integration for evaluating the volume of the entire solid.

### Worked Examples of Volumes by Cross Sections

The base of a solid is formed by the area bounded by $y = \cos x$ and $y = -\cos x$ for $\displaystyle 0 \le x \le \frac{\pi}{2}$. Vertical cross sections of the solid taken parallel to the $y$-axis are in the shape of isosceles triangles with the equal sides of length 1 unit. Find the volume of the solid.

\begin{aligned} \displaystyle \text{base of the triangle } &= 2 \cos x \\ \text{height of the triangle } &= \sqrt{1- \cos^2 x} \\ \delta V &= \frac{1}{2} \times 2 \cos x \times \sqrt{1- \cos^2 x} \times \delta x \\ &= \cos x \times \sin x \delta x \\ &= \frac{1}{2} \sin 2x \delta x \\ V &= \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sin 2x dx \\ &= \frac{1}{2}\Big[-\frac{1}{2} \cos 2x \Big]_{0}^{\frac{\pi}{2}} \\ &= -\frac{1}{4}\Big[\cos 2x \Big]_{0}^{\frac{\pi}{2}} \\ &= -\frac{1}{4}\Big[\cos \pi – \cos 0\Big] \\ &= -\frac{1}{4}\Big[-1-1] \\ &= \frac{1}{2} \\ \end{aligned} \\ 