Introduction to Volumes with Rectangular Cross Sections

When dealing with volumes of solids with rectangular cross sections, we use the concept of integral calculus to compute the volume. This method is particularly useful when the cross-sectional area changes along the length of the solid.

#Basic Concept

To find the volume of a solid with a rectangular cross section, we use the following approach:

#Step-by-Step Process

Determine the Cross-Sectional Area Function

If the area of the cross section of a solid is given by $A(x)$ and $A(x)$ is continuous on $\left[a, b\right]$ , we can integrate this area function over the interval to find the volume of the solid.

$\text{Volume} = \int_{a}^{b} A(x) , dx$
Create the Cross-Sectional Area Function

Often, you may need to derive $A(x)$ based on the information provided in the problem. This function might depend on the values of another function (or functions) given in the question.

#Worked Example

Let's go through an example to understand this concept better.

#Problem Statement

Let $R$ be the region enclosed by the graph of $f(x) = 54 - 2x^3$ , the $x$ - and $y$ -axes, and the vertical line $x=2$ . $R$ is the base of a solid. For the solid, at each $x$ , the cross section perpendicular to the $x$ -axis is a rectangle with height $h(x) = 1 + x$ . Find the volume of the solid.

#Solution

Identify the Cross-Sectional Area Function

The cross-sectional area at each $x$ is given by the product of the height of the rectangle and the value of the function $f(x)$ :

$A(x) = f(x) \cdot h(x) = (54 - 2x^3)(1 + x)$
Expand the Area Function

$A(x) = (54 - 2x^3)(1 + x) = 54 + 54x - 2x^3 - 2x^4$
Set Up the Integral

To find the volume, we integrate $A(x)$ from $x=0$ to $x=2$ :

$\text{Volume} = \int_{0}^{2} (54 + 54x - 2x^3 - 2x^4) , dx$
Compute the Integral

$\text{Volume} = \left[ 54x + 27x^2 - \frac{1}{2}x^4 - \frac{2}{5}x^5 \right]_{0}^{2}$
Evaluate the Integral

$\text{Volume} = \left(54(2) + 27(2)^2 - \frac{1}{2}(2)^4 - \frac{2}{5}(2)^5 \right) - \left(0\right)$ $\text{Volume} = \left(108 + 108 - 8 - \frac{64}{5} \right)$ $\text{Volume} = \frac{976}{5}$ $\text{Volume} = 195.2 , \text{units}^3$

Thus, the volume of the solid is 195.2 cubic units.

#Practice Questions

#

Practice Question

Question 1

Let

R

be the region enclosed by the graph of

g(x) = x^2

, the

x

-axis, and the vertical lines

x=0

and

x=3

.

R

is the base of a solid where each cross section perpendicular to the

x

-axis is a rectangle with height

h(x) = 2x

. Find the volume of the solid.

#

Practice Question

Question 2

A solid has a base in the shape of a semicircle with radius 3. Each cross section perpendicular to the diameter is a rectangle with height equal to the distance from the center of the semicircle. Find the volume of the solid.

#Glossary

Cross-Sectional Area Function ( $A(x)$ ): A function that describes the area of a cross section of a solid at a particular value of $x$ .
Integral: A mathematical concept that represents the area under a curve and is used to calculate volumes, among other things.

#Summary and Key Takeaways

To find the volume of a solid with rectangular cross sections, determine the area function $A(x)$ and integrate it over the given interval.
The formula for volume is $\text{Volume} = \int_{a}^{b} A(x) , dx$ .
Ensure that the cross-sectional area function is continuous over the interval.

#Key Takeaways

Understand the problem statement: Identify the limits of integration and the functions involved.
Create the cross-sectional area function: Use the given information to express $A(x)$ .
Set up and evaluate the integral: Integrate $A(x)$ over the specified interval to find the volume.