BackVolumes of Solids with Known Cross-Sections: Disc and Washer Methods
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Volumes of Solids with Known Cross-Sections
Introduction
In calculus, one important application of integration is finding the volume of solids whose cross-sections are known. This topic covers the methods for computing such volumes, including the disc and washer methods, and provides examples for each case.
Solids with Perpendicular Cross-Sections
General Approach
Cross-section: The intersection of a solid with a plane perpendicular to an axis (usually the x-axis).
Area of cross-section: If the cross-section at position has area , then the volume of the solid from to is:
Method: Divide the solid into thin slices, approximate each slice's volume, and sum using integration.
Example: Cone with Circular Cross-Sections
Given: Cone at the origin, length $5.
Cross-section at : Circle with radius .
Area:
Volume estimate (using slices):
Exact volume:
Solids with Square and Semi-Circular Cross-Sections
Squares as Cross-Sections
Region : Bounded by and the x-axis.
Cross-section: Squares perpendicular to the x-axis.
Area of square:
Volume:
Solution: Expand and integrate:
Semi-Circles as Cross-Sections
Region : Same as above, but cross-sections are semi-circles.
Diameter:
Radius:
Area of semi-circle:
Volume:
Disc and Washer Methods
Disc Method
The disc method is used when the cross-sections perpendicular to the axis of revolution are discs.
Area of disc:
Volume:
Example: Region bounded by , , and the x-axis, revolved around the x-axis.
Area:
Volume:
Washer Method
The washer method is used when the cross-sections are washers (discs with holes), i.e., the region is revolved around an axis and has both an outer and inner radius.
Area of washer:
Volume:
Example: Region bounded by , , , revolved around the y-axis.
Outer radius:
Inner radius:
Volume:
Theorems: Disc and Washer Methods
Disc Method Theorem
If a solid has cross-sections perpendicular to the x-axis that are discs with radius , then its volume is:
Washer Method Theorem
If a solid has cross-sections perpendicular to the x-axis that are washers with outer radius and inner radius , then its volume is:
Summary Table: Disc vs. Washer Method
Method | Cross-Section Shape | Area Formula | Volume Formula |
|---|---|---|---|
Disc | Disc (no hole) | ||
Washer | Disc with hole (washer) |
Key Points
Disc method is used when the solid is generated by revolving a region around an axis and the cross-sections are solid discs.
Washer method is used when the solid is generated by revolving a region around an axis and the cross-sections are washers (discs with holes).
Always express the area of the cross-section in terms of (or ) before integrating.
Set up the limits of integration according to the bounds of the region.
Additional info:
These methods are foundational for more advanced topics in calculus, such as finding volumes of revolution and applications in physics and engineering.
