Find the volume of the solid obtained by rotating the region enclosed by Note: Y

Question

Find the volume of the solid obtained by rotating the region enclosed by

Note: You can earn 5% for the upper limit of integration, 5% for the lower limit of integration, 40% for the integrand, and 50% for the finding the volume. If you find the correct volume, you will get full credit no matter what your other answers are.

Using the washer method, set up the integral.

Using the method of cylindrical shells, set up the integral.

Choose either integral to find the volume.

Volume =

Volume =

  =   Find the volume of the solid obtained by rotating the region enclosed by y = 1/x3, y = 0, x = 2, and x = 6, about the line y = -5 using the method of disks or washers. Note: You can earn 5% for the upper limit of integration, 5% for the lower limit of integration, 40% for the integrand, and 50% for the finding the volume. If you find the correct volume, you will get full credit no matter what your other answers are. The volume of the solid obtained by rotating the region bounded by y = x2, and y = 3x about the line x = 3 can be computed using either the washer method or the method of cylindrical shells. Answer the following questions. Using the washer method, set up the integral. v = a = b = Using the method of cylindrical shells, set up the integral v = c = d =

Explanation / Answer

e.g

Thickness of "washer" = dy
Inner radius = y^3, outer radius = 5y
Area of washer = pi [25y^2 - y^6]
Limits on y are 0 and sqrt(5).

Before integrating I'm going to estimate the volume using 2nd Thm of Pappus. Right extremity is around 11.2. In fooplot.com, the area appears to be about 5 to 6, very roughly. The x-coordinate of the centroid is around 4 to 5, so the distance the centroid travels in the revolution is about 28. Hence the volume should come out on the order of 5.5*28 = 154.

Integrate from y = 0 to y = sqrt(5) the quantity
pi (25y^2 - y^6) dy
= pi [(25/3)y^3 - (1/7)y^7] to be evaluated at y = sqrt(5)
= pi*sqrt(5) [ 125/3 - 125/7 ]
= pi*5^(3.5)*(4/21) = about 167.

Seems close enough to the estimated answer.
Most conventional way to express it, I suppose, is
(500/21) pi sqrt(5)

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