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6.2: VolumesDiana Pell
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Exercise 1. Show that the volume of a sphere of radius r is V = 43πr
3.
Exercise 2. Find the volume of the solid obtained by rotating aboutthe x-axis the region under the curve y =
√x from 0 to 1.
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Exercise 3. Find the volume of the solid obtained by rotating the re-gion bounded by y = x3, y = 8, and x = 0 about the y-axis.
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Exercise 4. The region R enclosed by the curves y = x and y = x2 isrotated about the x-axis. Find the volume of the resulting solid.
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Exercise 5. Find the volume of the solid obtained by rotating the re-gion R enclosed by the curves y = x and y = x2 about the line y = 2.
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Solids of Revolution
V =
∫ b
a
A(x) dx or V =
∫ d
c
A(y) dy
To find A(x) or A(y), use a disk or a washer as the cross-section.
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Exercise 6. Find the volume of the solid obtained by rotating the re-gion R enclosed by the curves y = x and y = x2 about the line x = −1.
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Volume of solids that are not solids of revolution
Exercise 7. Figure below shows a solid with a circular base of radius1. Parallel cross-sections perpendicular to the base are equilateral tri-angles. Find the volume of the solid.
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Exercise 8. Find the volume of a pyramid whose base is a square withside L and whose height is h.
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