6.2: Volumes - websites.rcc.eduwebsites.rcc.edu/pell/files/2018/03/6.2-Volumes.pdf2018/03/06  · 6.2: Volumes Diana Pell 1 Exercise 1. Show that the volume of a sphere of radius r

6.2: VolumesDiana Pell

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Exercise 1. Show that the volume of a sphere of radius r is V = 43πr

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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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