Calculus

a) Find the volume formed by rotating the region enclosed by x = 6y and y^3 = x with y greater than, equal to 0 about the y-axis.

b) Find the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, and y = 0 about the x-axis.

c) Find the volume of the solid obtained by rotating the region bounded by y = 1/x^6, y = 0, x = 3, and x = 9, about the x-axis.

Thanks for your trouble.

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  1. I'll do (a)

    First, determine where the curves intersect:
    6y = y^3 at the point (6√6,√6)

    You can do this using discs or shells. Using discs, or washers, you have a stack of washers. The area of a disc of outer radius R and inner hole of radis r is

    m(R^2 - r^2)
    Since 6y > y^3 on the interval desired

    The thickness of each disc is dy. So, the volume of the stack of discs is the integral

    π Int(R^2 - r^2 dy)[0,√6]
    = π Int(36y^2 - (y^3)^2) dy)[0,√6]
    = π Int(36y^2 - y^6 dy)[0,√6]
    = π (12y^3 - 1/7 y^7)[0,√6]
    = π (12*6√6 - 1/7 * 216√6)
    = 36π√6(2 - 6/7)
    = 36*8π√6/7

    Or, if you want to calculate using shells, the volume of a shell is 2πrh

    r = x, h = x^(1/3) - x/6

    So, the volume of a ring of shells is

    2π Int(x(x^1/3 - x/6) dx )[0,63/2]
    = 2π Int(x(x^1/3 - x/6) dx )[0,63/2]
    = 2π Int(x4/3 - x2/6) dx )[0,63/2]
    = 2π(3/7 x7/3 - x3/18)[0,63/2]
    = 2π(67/2 - 69/2/18)
    = 2π*216√6(3/7 - 1/3)
    = π*216√6(4/21)
    = 36*8π√6/7

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  2. Thank you!

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