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Which answer is right? Please check my work.

  1. Feb 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Use polar coordinates to find the volume of the solid where T is the region that lies under the plane 3x+4y+z=12, above the xy-plane, and inside the cylinder x^2+y^2=2x.

    2. Relevant equations
    None.

    3. The attempt at a solution
    Here's my work:
    x^2+y^2=2x
    x^2-2x+y^2=0
    x^2-2x+1+y^2=1
    (x-1)^2+y^2=1
    (r*cos(theta)-1)^2+(r*sin(theta))^2=1
    r^2(cos(theta))^2-2r*cos(theta)+1+r^2(sin(theta))^2=1
    r^2-2r*cos(theta)=0
    factor
    r(r-2*cos(theta))=0
    r-2cos(theta)=0
    r=2cos(theta)
    V=r dz dr d(theta) from 0 to 2pi, from 0 to 2cos(theta), from 0 to 12-3r(cos(theta))-4r(sin(theta))
    =18pi
    But the answer in the book is 9pi. Which answer is right?
     
  2. jcsd
  3. Feb 28, 2015 #2

    LCKurtz

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    The book is. Check your ##\theta## limits.
     
  4. Feb 28, 2015 #3
    So how do I find the theta limits? Is it from -2pi to 2pi?
     
  5. Feb 28, 2015 #4

    LCKurtz

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    Plot the graph of the polar equation circle and see what ##\theta## you need to get the circle.
     
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