Table of Contents

1. Introduction
2. Basic Concept
3. Volume Calculation with Square Cross Sections
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

Introduction

In this section, we will explore how to calculate the volume of a solid whose cross sections are squares. This is a fundamental concept in integral calculus often used in both mathematical problems and real-world applications.

Basic Concept

To find the volume of a solid with a square cross section, we use the integral of the area of the cross section along the axis of the solid.

Volume Calculation with Square Cross Sections

Often, you will need to create the cross-sectional area function $A(x)$ based on the problem's context. Let's consider a specific example to illustrate this process.

Example: Volume of a Square-Based Right Pyramid

To calculate the volume of a square-based right pyramid with height $h$ and base side length 2a, follow these steps:

1. Understand the Geometry:

   • Consider a side view of the pyramid along the $x$-axis.

2. Equation for the Height:

   • The height $h$ has the equation $y = a - \frac{a}{h}x$.

3. Cross-Sectional Area:

   • At each value of $x$, the cross section of the pyramid is a square with side length 2 \left(a - \frac{a}{h}x\right) and area: $$A(x) = \left(2 \left(a - \frac{a}{h}x\right)\right)^2 = 4a^2 \left(1 - \frac{x}{h}\right)^2$$

4. Integrate to Find Volume: $$\text{Volume} = \int_{0}^{h} 4a^2 \left(1 - \frac{x}{h}\right)^2 , dx$$ $$= 4a^2 \int_{0}^{h} \left(1 - \frac{2x}{h} + \frac{x^2}{h^2}\right) , dx$$ $$= 4a^2 \left[ x - \frac{x^2}{h} + \frac{x^3}{3h^2} \right]_{0}^{h}$$ $$= 4a^2 \left( h - \frac{h^2}{h} + \frac{h^3}{3h^2} \right)$$ $$= 4a^2 \left( h - h + \frac{h}{3} \right)$$ $$= \frac{4}{3}a^2h$$

Worked Example

Problem Statement

Consider a solid with square cross sections perpendicular to the $x$-axis, bounded by the $x$- and $y$-axes, and the vertical line $x=3$. The cross-sectional area is given by $A(x) = \left(1 + e^{-x}\right)^2$. Find the volume of the solid.

Solution

1. Set Up the Integral: $$\text{Volume} = \int_{0}^{3} \left(1 + e^{-x}\right)^2 , dx$$

2. Expand the Integrand: $$\left(1 + e^{-x}\right)^2 = 1 + 2e^{-x} + e^{-2x}$$

3. Integrate: $$\text{Volume} = \int_{0}^{3} \left(1 + 2e^{-x} + e^{-2x}\right) , dx$$ $$= \left[x - 2e^{-x} - \frac{1}{2}e^{-2x}\right]_{0}^{3}$$ $$= \left(3 - 2e^{-3} - \frac{1}{2}e^{-6}\right) - \left(0 - 2e^{0} - \frac{1}{2}e^{0}\right)$$ $$= 3 - 2e^{-3} - \frac{1}{2}e^{-6} + 2 + \frac{1}{2}$$ $$= \frac{11}{2} - 2e^{-3} - \frac{1}{2}e^{-6}$$ $$\approx 5.399$$ (to 3 decimal places)

Practice Questions

Practice Question

1. Calculate the volume of a solid with square cross sections perpendicular to the $x$-axis, bounded by $x = 0$ and $x = 2$, where the side length of each square cross section is 1 + x.

Practice Question

2. Find the volume of a solid with square cross sections perpendicular to the $y$-axis, bounded by $y = 0$ and $y = 4$, where the side length of each square cross section is 2y.

Glossary

• Cross Section: A surface or shape exposed by making a straight cut through a solid.
• Integral: A mathematical object that represents the area under a curve.
• Volume: The amount of three-dimensional space occupied by an object.

Summary and Key Takeaways

• The volume of a solid with square cross sections can be found using the integral of the area function $A(x)$.
• Ensure $A(x)$ is continuous over the interval $[a, b]$.
• Use the formula: $$\text{Volume} = \int_{a}^{b} A(x) , dx$$
• Practice setting up and solving integrals based on given functions and geometric contexts.

Key Takeaways

• Use integration to find volumes with irregular shapes.
• Square cross sections simplify the process as the area function is squared.
• Practice with various functions to become proficient in setting up and solving integrals.

Feel confident and practice regularly. Good luck with your studies!