Volumes with Cross Sections

Sarah Miller

5 min read

Next Topic - Rectangles as Cross Sections

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Study Guide Overview

This study guide covers calculating volumes of solids with square cross sections using integration. It introduces the basic concept of using the integral of the area function A(x), provides an example of a square-based pyramid, and shows a worked example with a given A(x). The guide also includes practice questions, a glossary, 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.

Key Concept

If the area of the cross section of a solid is given by $A(x)$ and $A(x)$ is continuous on $[a, b]$ , then the volume of the corresponding solid from $x = a$ to $x = b$ is given by: $\text{Volume} = \int_{a}^{b} A(x) , dx$

#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...

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Previous Topic - Volumes from Areas of Known Cross Sections Next Topic - Rectangles as Cross Sections

Question 1 of 9

What is the formula to calculate the volume of a solid with a known cross-sectional area $A(x)$ from $x=a$ to $x=b$ ?

$\int_{a}^{b} A(x) , dx$

$A(x) \cdot (b-a)$

$\frac{A(b) - A(a)}{b-a}$

$\int_{a}^{b} A'(x) , dx$