Volumes with Cross Sections

Sarah Miller
5 min read
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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.
Table of Contents
- Introduction
- Volume of Solids with Semicircular Cross Sections
- Worked Example
- Practice Questions
- Glossary
- Summary and Key Takeaways
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 and is continuous on , then the volume of the corresponding solid from to is:
Creating the Cross-Sectional Area Function
You may need to create the cross-sectional area function based on the information provided in the problem. Often, may depend on the values of another function (or functions) given in the question.
Area of a Semicircle
The area of a circle is . Therefore, the area of a semicircle with radius is:
Worked Example
Let be the triangular region with vertices , , and . is the base of a solid. For the solid, at each , the cross-section perpendicular to the -axis is a semicircle. Find the volume of the solid.
Solution Steps
-
Find the Equation of the Line:
- The line passes through points and .
- Slope (gradient) is:
- Equation of the line:
-
Determine the Radius:
- The diameter of each semicircle is
2 - \frac{1}{2}x
. - Therefore, the radius is:
- The diameter of each semicircle is
-
Find the Cross-Sectional Area :
- Area of a semicircle:
- Simplify:
-
Calculate the Volume:
- Integrate from 0 to 4:
- Evaluate the integral:
- Substitute and calculate:
- Numerical value:
Practice Questions
Practice Question
- Find the volume of a solid with a base region defined by , from to , with cross sections perpendicular to the -axis that are semicircles.
Practice Question
- The base of a solid is a rectangle with length 6 and width 2. Cross sections perpendicular to the length are semicircles. Calculate the volume of the solid.
Practice Question
- A solid has a base defined by the region and . Cross sections perpendicular to the -axis are semicircles. Determine the volume of the solid.
Glossary
- Cross-Sectional Area (): The area of the shape obtained by cutting through a solid perpendicular to a given axis at position .
- Integral: A mathematical operation that calculates the area under a curve.
- Semicircle: A half of a circle.
- Radius: The distance from the center of a circle to any point on its circumference.
- Diameter: The distance across a circle through its center, equal to twice the radius.
Summary and Key Takeaways
Summary
- To find the volume of a solid with semicircular cross sections, use the integral of the area function .
- The area of a semicircle is .
- The integral is evaluated over the range of values that define the solid.
Key Takeaways
- Understand the formula:
- Identify the radius: Based on the geometry of the problem.
- Set up the integral: Use the correct limits and area function.
- Evaluate the integral: Carefully substitute and simplify to find the volume.
Always check your limits of integration and ensure your area function is correctly defined.

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