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Questions and Answers
What is the radius of the shell?
What is the radius of the shell?
What is the height of the shell?
What is the height of the shell?
What is the integral used to find the volume?
What is the integral used to find the volume?
What is the volume of the solid?
What is the volume of the solid?
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What is the alternate method used to find the volume?
What is the alternate method used to find the volume?
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What is the integral used in the washer method?
What is the integral used in the washer method?
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What is the purpose of identifying the limits of integration?
What is the purpose of identifying the limits of integration?
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What is the advantage of using the washer method?
What is the advantage of using the washer method?
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What is the disadvantage of using the shell method?
What is the disadvantage of using the shell method?
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What is the purpose of finding the volume of a solid?
What is the purpose of finding the volume of a solid?
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Study Notes
Volumes of Solids of Revolution
- Solids of revolution are solids whose shapes can be generated by revolving plane regions about axes.
- The method of slicing is commonly used to find the volumes of solids of revolution.
- The disk method, washer method, and cylindrical shell method can be used to find the volumes of solids of revolution.
Disk Method
- If a region bounded by a curve y = f(x), the x-axis, x = a, and x = b is revolved about the x-axis, a solid is generated.
- The typical cross-section of the solid perpendicular to the axis of revolution is a disk of radius f(x) and area A(x) = π[f(x)]².
- The solid's volume is the integral of A from x = a to x = b.
Exercises
- Find the volume of a cap of height h formed from a sphere of radius r.
- Find the volume of a right pyramid with a square base of side and height.
- Calculate the volume of the solid generated by revolving the plane region bounded by y = x, x = 4, and y = 1/2 about the x-axis.
- Compute the volume of the solid generated by revolving the plane region bounded by y = x², y = 9, and x = 0 about the x-axis.
Examples
- The base of a solid is an ellipse b²x² + a²y² = a²b². Each cross-section perpendicular to the x-axis is a square with ends of a side on the ellipse.
- A solid has a circular base of radius 2. Parallel cross-sections perpendicular to its base are equilateral triangles.
- The radius of a hemispherical vat is 5 ft, and it contains a liquid to a depth of 4 ft. Find the volume of the liquid.
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Description
Test your understanding of volumes of solids of revolution with these 10 questions. Calculate volumes of caps, pyramids, and solids generated by revolving plane regions.
