Post a New Question

Calculus

posted by .

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.

  • Calculus -

    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

  • Calculus -

    Thank you!

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. Calculus

    1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y¡Ý0 about the y-axis. 2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=18x-6x^2 …
  2. calculus edit

    1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y is greater than or equal to 0 about the y-axis. 2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the …
  3. Calculus

    Find the volume of the solid obtained by rotating the region enclosed by the graphs of y=15-x, y=3x+11 and x=0 about the y-axis.
  4. Calculus

    This problem set is ridiculously hard. I know how to find the volume of a solid (integrate using the limits of integration), but these questions seem more advanced than usual. Please help and thanks in advance! 1. Find the volume of …
  5. calculus

    Sketch the region bounded by the curves y = x^2, y = x^4. 1) Find the area of the region enclosed by the two curves; 2) Find the volume of the solid obtained by rotating the above region about the x-axis; 3) Find the volume of the …
  6. calculus

    Find the volume of the solid formed by rotating the region enclosed by y=e^(2x)+1 y=0 x=0 x=0.9 about the y-axis.
  7. CALCULUS MAJOR HELP!!!!!!

    Find the volume of the solid obtained by rotating the region bounded y = 16 x and y = 2 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded about the x-axis by y=4x^2, x =1, and y = 0 Find the volume …
  8. Calculus volume stuff

    Find the volume of the solid obtained by rotating the region bounded y = 16 x and y = 2 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded about the x-axis by y=4x^2, x =1, and y = 0 Find the volume …
  9. Calculus I don't understand

    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = 10 x and y = 5 x^2 about y =0 Find the volume of the solid obtained by rotating the region bounded by y=8 x^2, x …
  10. Calculus

    Find the volume of the solid obtained by rotating the region enclosed by the graphs of y=18-x, y=3x-6 and x=0 about the y-axis V=

More Similar Questions

Post a New Question