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Volumes with Cross Sections

Sarah Miller

Sarah Miller

5 min read

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Study Guide Overview

This study guide covers calculating volumes of solids with known cross-sectional areas using definite integrals. It explains the key concepts, provides a step-by-step approach, and offers worked examples and practice questions. The guide also includes a glossary and emphasizes the importance of the cross-sectional area function, integral setup, and correct units.

Volumes from Areas of Known Cross Sections

Table of Contents

  1. Introduction
  2. Finding the Volume of a Solid with Known Cross Sections
  3. Worked Example
  4. Practice Questions
  5. Glossary
  6. 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 xx, 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)A(x) over the interval [a,b][a, b] can be found using the integral: Volume=∫abA(x),dx\text{Volume} = \int_{a}^{b} A(x) , dx

Steps to Find the Volume

  1. Identify the Cross-Sectional Area Function: The cross-sectional area A(x)A(x) must be given or derivable.
  2. Set Up the Integral: Use the limits of integration aa and bb which correspond to the range over which the solid extends.
  3. 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)=1x+1A(x) = \frac{1}{x+1} for $0 \le x \le 3$. Find the volume of the solid.

Solution:

  1. Identify the Cross-Sectional Area Function: A(x)=1x+1A(x) = \frac{1}{x+1}
  2. Set Up the Integral: Volume=∫031x+1,dx\text{Volume} = \int_{0}^{3} \frac{1}{x+1} , dx
  3. Evaluate the Integral: Volume=[ln⁡(x+1)]03=ln⁡(4)−ln⁡(1)=ln⁡(4)−0=ln⁡(4)\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)\ln(4) cubic units.

Understanding the Integral

The method of finding volumes using integrals involves calculating the **accumulation of change**. Here, A(x) dxA(x) \, dx represents a small volume element, and the integral sums these elements from aa to bb.

Worked Example

**Problem**: The area, in square feet, of the horizontal cross section of a water tank at height hh feet is modeled by the function ff given by f(h)=50ehf(h) = \frac{50}{e^h}. The tank has a height of 10 feet. Find the volume of the tank.

Solution:

  1. Identify the Cross-Sectional Area Function: A(h)=50ehA(h) = \frac{50}{e^h}
  2. Set Up the Integral: Volume=∫01050eh,dh\text{Volume} = \int_{0}^{10} \frac{50}{e^h} , dh
  3. Evaluate the Integral: Volume=50∫010e−h,dh=50[−e−h]010=50(−e−10−(−e0))=50(1−e−10)≈50(1−0.0000454)≈49.998\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
  1. The cross-sectional area of a solid is given by A(x)=x2+1A(x) = x^2 + 1 for 0 \le x \le 2. Find the volume of the solid.
Practice Question
  1. The area of a vertical cross-section of a solid is A(y)=sin⁡(y)A(y) = \sin(y) for 0 \le y \le \pi. Calculate the volume of the solid.
Practice Question
  1. A solid has cross-sectional area A(z)=3z2+2A(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: Volume=∫abA(x),dx\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.

Previous Topic - Multiple AreasNext Topic - Squares as Cross Sections

Question 1 of 12

Alright, let's start with an easy one! 🚀 If the cross-sectional area of a solid is given by A(x)=2xA(x) = 2x and we want to find the volume between x=0x = 0 and x=1x = 1, which integral should we evaluate?

∫012x,dx\int_{0}^{1} 2x , dx

∫01x2,dx\int_{0}^{1} x^2 , dx

∫012,dx\int_{0}^{1} 2 , dx

∫022x,dx\int_{0}^{2} 2x , dx