the solid generated by rotating the bounded region about a given axis. Find the

Question

the solid generated by rotating the bounded region about a given axis. Find the volume of such solids, as well as solids of known cross sections, by applying the definition of volume. Problems 1. Find the volume of the solid obtained by rotating the region bounded by r 2vy, r 0, and y 9 about the y-axis. 2. Find the volume of the solid obtained by rotating the region bounded by r and r 2y about the y-axis. 3. Find the volume of the solid obtained by rotating the region bounded by y cos(r) and y Sin (5) between r a -T and r T around the line y 1. 4. Find the volume of the solid obtained by rotating the region bounded by y cos(z) and y sin between r -r and r 3T around the line 5. Find the volume of the solid obtained by rotating the region bounded by y A+ cos r and y 1 between r T/3 and r T/3 about the line y 1

Explanation / Answer

1) use the fact that a volume of revolution in this case can be done by using the disc method.

V = pi( integral from 0 to 9 of x^2 dy).

Now, the radius of the revolution is x .

x = 2sqrt(y), so x^2 = 4y. The volume becomes

V = pi (integral from 0 to 9 of 4y dy)
= pi 2y^2 from 0 to 9
= pi (2(9)^2 - 2(0)^2)
= 162 pi.

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