Cross Sections: Triangles and Semicircles Cross Sections: Triangles and Semicircles

Introduction

The area of a square cross-section is

A(x) = f(x)^2


The volume of a rectangular cross-section is
A(x) = f(x) \cdot g(x)


The area of an equilateral triangular cross-section, where the side-length is equal to
f(x)
, is
\begin{align*} A(x) &= \frac{1}{2} b \cdot h \\\\ &= \frac{1}{2} b \cdot \frac{\sqrt{3}}{2} b \\\\ &= \frac{\sqrt{3}}{4} b^2 \\\\ &= \frac{\sqrt{3}}{4} f(x)^2 \end{align*}
For example, a solid with a triangle cross-section of
f(x)=x^2
.


The volume of a semicircular cross-section, where the diameter is equal to
f(x)
, is
\begin{align*} A(x) &= \frac{1}{2} \pi r^2 \\\\ &= \frac{1}{2} \pi \left(\frac{d}{2}\right)^2 \\\\ &= \frac{\pi}{8} d^2 \\\\ &= \frac{\pi}{8} f(x)^2 \end{align*}
For example, a solid with a semicircular cross-section of
f(x)=x^2
.

Examples

Example 1: Equilateral Triangle Cross Sections
Find the volume of the solid whose base is bounded by

y = \sqrt{x}
on
[0,4]
, with equilateral triangle cross sections perpendicular to the x-axis.

\begin{align*} \text{Volume} &= \int_{0}^{4} A \cdot d \\\\ &= \int_{0}^{4} \frac{\sqrt{3}}{4}f(x)^2 \cdot d \\\\ &= \int_{0}^{4} \frac{\sqrt{3}}{4} x \, dx \\\\ &= \frac{\sqrt{3}}{4} \left[ \frac{1}{2}x^2 \right]_0^{4} \\\\ &= \frac{\sqrt{3}}{8} (16 - 0) \\\\ &= 2\sqrt{3} \approx 3.464 \, \text{units}^3 \end{align*}
The volume is
2\sqrt{3}
cubic units.

Example 2: Semicircular Cross Sections
Find the volume of the solid bounded by
y = \sin(x)
and the x-axis on
[0, \pi]
, with semicircular cross sections perpendicular to the x-axis.

A(x)=\frac{\pi}{8} \sin^2(x)
.
\begin{align*} \text{Volume} &= \int_{0}^{\pi} A \cdot d \\\\ &= \int_{0}^{\pi} \frac{\pi}{8} f(x)^2 \, dx \\\\ &= \int_{0}^{\pi} \frac{\pi}{8} \sin^2(x) \, dx \\\\ &= \frac{\pi}{8} \int_{0}^{\pi} \frac{1 - \cos(2x)}{2} \, dx \\\\ &= \frac{\pi}{16} \left[ x - \frac{\sin(2x)}{2} \right]_0^{\pi} \\\\ &= \frac{\pi}{16} (\pi - 0) \\\\ &= \frac{\pi^2}{16} \approx 0.617 \, \text{units}^3 \end{align*}
The volume is
\pi^2/16
cubic units.