Calculus

Find the volume of the solid obtained by rotating the region bounded by the curves y=cos(x), y=0, x=0, and x=π/2 about the line y=1.

  1. 👍
  2. 👎
  3. 👁
  4. ℹ️
  5. 🚩
  1. radius of cylinder of skin thickness dr (r goes from 0 to 1)
    r = 1-cos x
    dr = sin x dx
    so circumference = 2 π (1-cos x)
    height of that cylinderical shell = x
    dV = 2 π(1-cos x) x dr
    = 2 π (1-cos x)x sin x dx
    = 2 π x sin x dx - 2 π x sin x cos x dx

    = 2 π [ sin x-x cos x]
    - 2 π/8 [ sin 2 x - 2 x cos 2x]
    (I used wolfram alpha integral - google it)
    evaluate at x = pi/2
    = 2 π [ 1]
    - 2 π/8 [ 0 + π ]

    = 2 π - π^2 /4
    now
    evaluate at x = 0
    = 2 π[ 0 ]
    - π/4 [ 0 ] handy
    so
    2 π - π^2 /4

    1. 👍
    2. 👎
    3. ℹ️
    4. 🚩
  2. lots of algebra here
    here is the integral site:
    http://www.wolframalpha.com/calculators/integral-calculator/

    1. 👍
    2. 👎
    3. ℹ️
    4. 🚩
  3. works for me. I did the integral directly, using

    ∫[0,1] 2π(1-y) arccos(y) dy

    and got the same answer.

    I like your technique of avoiding the arccos, though.

    1. 👍
    2. 👎
    3. ℹ️
    4. 🚩

Respond to this Question

First Name

Your Response

Similar Questions

  1. calculus

    Find the volume of the solid obtained by rotating the region bounded by the curves y = x^8, y = 1 about the line y = 5.

  2. Calculus

    1. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6. y = x, y = 0, y = 5, x = 6 2. Use the method of cylindrical shells to find the volume V generated by

  3. Ap calc

    Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. x + y = 3, x = 4 − (y − 1)^2

  4. calculus

    Consider the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 9 - 9x^2 , y = 0 Find the volume V of this solid. Sketch the region, the solid, and a typical disk or washer. Any help or tips

  1. Math

    Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 8 sin x, y = 8 cos x, 0 ≤ x ≤ π/4; about y = −1

  2. 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 =

  3. calculus

    Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. xy = 2, x = 0, y = 2, y = 4

  4. calculus

    1. Find the volume V obtained by rotating the region bounded by the curves about the given axis. y = sin(x), y = 0, π/2 ≤ x ≤ π; about the x−axis 2. Find the volume V obtained by rotating the region bounded by the curves

  1. calculus review please help!

    1) Find the area of the region bounded by the curves y=arcsin (x/4), y = 0, and x = 4 obtained by integrating with respect to y. Your work must include the definite integral and the antiderivative. 2)Set up, but do not evaluate,

  2. calculus help

    Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. x = 4(square root of (3y)), x = 0, y = 3; about the y-axis V=?????

  3. calculus

    1. Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln(5x), y = 1, y = 3, x = 0; about the y-axis 2. Use the method of cylindrical shells to find the volume V

  4. calc

    Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=0, y=cos(7x) , x=π/14, x=0 about the axis y=−8

View more similar questions or ask a new question.