Volumes with Cross Sections

Sarah Miller

5 min read

Next Topic - Disc Method Around the x-Axis

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Study Guide Overview

This guide covers calculating the volume of solids with semicircular cross sections using integration. It explains the core concept of integrating the cross-sectional area function, A(x), and the formula for the area of a semicircle. A worked example demonstrates finding A(x) from a given base region, setting up the definite integral, and evaluating it to determine the volume. Practice questions and a glossary of terms like radius, diameter, and integral are also included.

Introduction In this guide, we will explore how to find the volume of a solid with semicircular cross sections. This involves integrating the area of the cross-section along the axis of the solid.

#Volume of Solids with Semicircular Cross Sections

#Basic Concept

To find the volume of a solid with a given cross-sectional area:

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: $\mathrm{Volume} = \int_{a}^{b} A(x) , dx$

#Creating the Cross-Sectional Area Function

You may need to create the cross-sectional area function $A(x)$ based on the information provided in the problem...

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Question 1 of 9

🎉 What is the general formula to calculate the volume of a solid using cross-sectional areas?

$Volume = A(x) \cdot \Delta x$

$Volume = \int A(x) , dx$

$Volume = \frac{1}{2} \pi r^2$

$Volume = \pi r^2 h$