1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find the volume of the solid

  1. Nov 29, 2014 #1
    1. The problem statement, all variables and given/known data

    find the volume of teh solid based on the interior of the circle, r=cos(theta), and capped by the plane z=x.

    2. Relevant equations


    3. The attempt at a solution

    i have drawn out the circle of equation r=cos(theta). I think that since z=x and is above the region, we have to use the double integral over this circular region. and integrate the function f(x,y)=x which in polar coordinate for would be rcos(theta).

    so my train of thought is the following:

    (∫ dtheta )(∫ (rcos(theta))rdr
    where the limits of integration are for ∫ dtheta 0 to 2π
    for ∫ (rcos(theta)rdr are from 0 to rcos(theta)

    im not sure if my integrals are set up correctly any help regarding this problem would be very much appreciated. I have been stuck on this for a while
     
  2. jcsd
  3. Nov 29, 2014 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    It looks good except for your limits on the ##\theta## integral.
     
  4. Nov 29, 2014 #3
    So how would I go about finding the limits for the r integral? I also think that the function which I'm integrating is incorrect.
     
  5. Nov 29, 2014 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    I didn't notice the mistake in the limits on the ##r## integral as well.

    Suppose you wanted to calculate the area of the circle. What integral would you set up for that?
     
  6. Nov 29, 2014 #5
    to calculate area by double integral i would have to integrate over the constant 1; my bounds of integration would be from r=0 to r=cos(theta) and i would have the second integral from theta=-Pi/2 to theta=Pi/2 for the theta integral
     
  7. Nov 29, 2014 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Good. Now to get the volume, instead of integrating the function 1, you want to integrate the height of the solid as a function of ##r## and ##\theta##.
     
  8. Nov 29, 2014 #7
    So since the circle is capped be the plane, we have to integrate over the function f (x,y)=x which in polar coordinates is rcos (theta). I'm still not a 100% sure if this is correct or not.
     
  9. Nov 30, 2014 #8

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You're doing fine. What integral do you get?
     
  10. Nov 30, 2014 #9
    So I set it up the following way:
    The integral for theta went from -pi/2 to pi/2 and the integral for r went from 0 to cos (theta) the function which I integrated was rcos (theta) the result was pi/8

    Can anybody confirm this?
     
  11. Nov 30, 2014 #10

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Doesn't sound right, though I've not checked in detail. I meant, what integral expression do you get?
     
  12. Nov 30, 2014 #11
    my
    my integral expression was as follows: integral dtheta (bounds from -pi/2 to pi/2) integral rdr (bounds from r=0 and r=cos(theta) and the expression which i integrate is x. which i wrote as x=rcos(theta)
     
  13. Nov 30, 2014 #12

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Ok, that's correct, and I do get pi/8. I hadn't expected the 1/3 to get cancelled.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Find the volume of the solid
Loading...