Cross Sections: Squares and Rectangles Cross Sections: Squares and Rectangles
Introduction
The next four lessons are on finding the volumes of solids which are created by manipulating a function.
All of the solids have these characteristics in common:
The length of the solid is the length of the domain of the function.
The cross-section of the solid is derived from the function.
A cross-section of a solid is the shape formed by
slicing it perpendicular to the axis of its inputs.
For example, if we take a pyramid and slice it horizontally, the cross-section is a
triangle. No matter where we slice it, the cross-section is always a triangle.
For all of the solids in these lessons, we will be using the same method to find the volume:
Find the volume of every cross-section.
Sum the volumes of all the cross-sections.
This is another Riemann sum, but in three dimensions instead of two.
With Riemann sums for area, the formula was:
\begin{align*}
\text{Area} &= \sum_{i=1}^{n} w \cdot h \\
&= \sum_{i=1}^{n} f(x_i) \Delta x \\
&= \int_{a}^{b} f(x) \, dx
\end{align*}
dx
was the width of the rectangle.
f(x_i) was the height of the rectangle.
For volume, the formula is:
\begin{align*}
\text{Volume} &= \sum_{i=1}^{n} A \cdot d \\
&= \sum_{i=1}^{n} A(x_i) \Delta x \\
&= \int_{a}^{b} A(x) \, dx
\end{align*}
dx
is the width of the cross-section.
A(x_i)
is the area of the cross-section.
A
can be derived from
f
.
Square Cross-Sections
A volume with a square cross-section is one where
A(x) = f(x)^2
The height and depth of the cross-section are equal to the value of the function.
The formula for the volume of a solid with square cross-sections is:
\text{Volume} = \int_{a}^{b} f(x)^2 \, dx
For example, here is a solid with square cross-sections of
f(x)=x^2
. Hover over the sketch to see the cross-section.
Rectangular Cross-Sections
Rectangles are similar, but the depth is not necessarily the same as the height.
For a solid with rectangular cross-sections of height
f(x)
, width
dx
, and depth
g(x)
, the formula for the volume is
\begin{align*}
\text{Volume} &= \sum_{i=1}^{n} h \cdot d \cdot w \\
&= \sum_{i=1}^{n} f(x_i) \cdot g(x_i) \cdot \Delta x \\
&= \int_{a}^{b} f(x) \cdot g(x) \, dx
\end{align*}
For example, here is a solid with rectangular cross-sections of
f(x)=x
and
g(x)=x^2
.
Examples
Example 1: Square Cross Sections
Find the volume of the solid whose base is bounded by
y = x - x^2
on
[0,1]
.
\begin{align*}
\text{Volume} &= \int_{0}^{1} A \cdot d \\
&= \int_{0}^{1} (x - x^2)^2 \, dx \\
&= \int_{0}^{1} (x^2 - 2x^3 + x^4) \, dx \\
&= \left[ \frac{1}{3}x^3 - \frac{1}{2}x^4 + \frac{1}{5}x^5 \right]_0^1 \\
&= \frac{1}{3} - \frac{1}{2} + \frac{1}{5} = \frac{1}{30} \approx 0.033 \, \text{units}^3
\end{align*}
The volume of the solid is
1/30
cubic units.
Example 2: Rectangular Cross Sections
Find the volume of the solid bounded by
y = \sin(x)
and the x-axis
on
[0,\pi]
, with rectangular cross sections
perpendicular to the x-axis. The depth of each rectangle is half of the height.
\begin{align*}
\text{Volume} &= \int_{0}^{\pi} f(x) \cdot g(x) \, dx \\
&= \int_{0}^{\pi} \sin(x) \cdot \frac{1}{2}\sin(x) \, dx \\
&= \frac{1}{2} \int_{0}^{\pi} \sin^2(x) \, dx \\
&= \frac{1}{2} \int_{0}^{\pi} \frac{1 - \cos(2x)}{2} \, dx \\
&= \frac{1}{4} \left[ x - \frac{\sin(2x)}{2} \right]_0^{\pi} \\
&= \frac{1}{4} (\pi - 0) = \frac{\pi}{4} \approx 0.785 \, \text{units}^3
\end{align*}
The volume of the solid is
\pi/4
cubic units.