Volumes with Cross Sections

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

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 giv...

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Question 1 of 12

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

$\int_{0}^{1} 2x , dx$

$\int_{0}^{1} x^2 , dx$

$\int_{0}^{1} 2 , dx$

$\int_{0}^{2} 2x , dx$