#Volumes from Areas of Known Cross Sections

#Table of Contents

Introduction
Finding the Volume of a Solid with Known Cross Sections
Worked Example
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction

In this section, we will learn how to find the volume of a solid when the area of its cross sections is known. This concept is crucial in calculus and is widely used in various real-world applications.

#Finding the Volume of a Solid with Known Cross Sections

When the area of the cross section of a solid can be expressed as a function of $x$ , we can find the volume of the solid using the definite integral.

#Key Concepts

Key Concept

Definition: The volume of a solid with cross-sectional area $A(x)$ over the interval $[a, b]$ can be found using the integral: $\text{Volume} = \int_{a}^{b} A(x) , dx$

#Steps to Find the Volume

Identify the Cross-Sectional Area Function: The cross-sectional area $A(x)$ must be given or derivable.
Set Up the Integral: Use the limits of integration $a$ and $b$ which correspond to the range over which the solid extends.
Evaluate the Integral: Compute the definite integral to find the volume.

#Example

**Problem**: The cross-sectional area of a solid is given by

A(x) = \frac{1}{x+1}

for

0 \le x \le 3

. Find the volume of the solid.

Solution:

Identify the Cross-Sectional Area Function: $A(x) = \frac{1}{x+1}$
Set Up the Integral: $\text{Volume} = \int_{0}^{3} \frac{1}{x+1} , dx$
Evaluate the Integral: $\text{Volume} = \left[ \ln(x+1) \right]_{0}^{3} = \ln(4) - \ln(1) = \ln(4) - 0 = \ln(4)$ Thus, the volume of the solid is $\ln(4)$ cubic units.

#Understanding the Integral

The method of finding volumes using integrals involves calculating the **accumulation of change**. Here,

A(x) \, dx

represents a small volume element, and the integral sums these elements from

a

to

b

.

#Worked Example

**Problem**: The area, in square feet, of the horizontal cross section of a water tank at height

h

feet is modeled by the function

f

given by

f(h) = \frac{50}{e^h}

. The tank has a height of 10 feet. Find the volume of the tank.

Solution:

Identify the Cross-Sectional Area Function: $A(h) = \frac{50}{e^h}$
Set Up the Integral: $\text{Volume} = \int_{0}^{10} \frac{50}{e^h} , dh$
Evaluate the Integral: $\begin{aligned} \text{Volume} &= 50 \int_{0}^{10} e^{-h} , dh \\ &= 50 \left[ -e^{-h} \right]_{0}^{10} \\ &= 50 \left( -e^{-10} - (-e^{0}) \right) \\ &= 50 \left(1 - e^{-10}\right) \\ &\approx 50 \left(1 - 0.0000454\right) \\ &\approx 49.998 \end{aligned}$ Thus, the volume of the tank is approximately 49.998 cubic feet (to 3 decimal places).

Units: Area is in square feet, and height is in feet, so the volume is in cubic feet.

#Practice Questions

Practice Question

The cross-sectional area of a solid is given by $A(x) = x^2 + 1$ for $0 \le x \le 2$ . Find the volume of the solid.

Practice Question

The area of a vertical cross-section of a solid is $A(y) = \sin(y)$ for $0 \le y \le \pi$ . Calculate the volume of the solid.

Practice Question

A solid has cross-sectional area $A(z) = 3z^2 + 2$ for $1 \le z \le 4$ . Determine the volume of the solid.

#Glossary

Cross Section: A shape obtained by cutting a solid perpendicular to a given axis.
Definite Integral: The integral of a function over a specified interval, giving the accumulation of quantities.
Volume: The amount of space occupied by a three-dimensional object, measured in cubic units.

#Summary and Key Takeaways

To find the volume of a solid with known cross-sectional areas, we use the integral of the cross-sectional area function over the given interval.
The volume formula is: $\text{Volume} = \int_{a}^{b} A(x) , dx$
Evaluate the integral carefully and ensure units are consistent.

#Key Takeaways

Understand the cross-sectional area function.
Set up and evaluate the integral correctly.
Include units in the final answer.

Exam Tip

Always double-check the limits of integration and the cross-sectional area function for accuracy.

By mastering these concepts, you can effectively find the volumes of solids with known cross-sectional areas, a skill valuable in both academic and real-world contexts.