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.

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

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