Results 1 to 2 of 2

Math Help - Integrating a circle with rectangular elements

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    30

    Integrating a circle with rectangular elements

    Imagine a circle where its center is at the origin with radius r. Now suppose I make infinitesimally thin rectangles only at the first quadrant, the rectangles length>width, from that one can see that I'm just trying to find a fourth of the total area of that circle. In terms of why, I have set up my integration problem to be: the integral from 0 to 4 of sqrt(r^2-x^2)dx times 4(since I'm trying to find the whole area, not just a fourth of the area). From that I'm just obviously trying to make the answer end up being (pi)r^2, the equation of a circle, but I can't figure out how to evaluate the integral from 0 to 4 of sqrt(r^2-x^2)dx times 4, unless its wrong... Anyways if it were right, how would I even do the integral, please show me your work, thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by maximade View Post
    Imagine a circle where its center is at the origin with radius r. Now suppose I make infinitesimally thin rectangles only at the first quadrant, the rectangles length>width, from that one can see that I'm just trying to find a fourth of the total area of that circle. In terms of why, I have set up my integration problem to be: the integral from 0 to 4 of sqrt(r^2-x^2)dx times 4(since I'm trying to find the whole area, not just a fourth of the area). From that I'm just obviously trying to make the answer end up being (pi)r^2, the equation of a circle, but I can't figure out how to evaluate the integral from 0 to 4 of sqrt(r^2-x^2)dx times 4, unless its wrong... Anyways if it were right, how would I even do the integral, please show me your work, thank you.
    \int_{0}^{r}\sqrt{r^2-x^2}dx

    let x=r\sin(t) \implies dx=r\cos(t)dt

    \int_{0}^{\frac{\pi}{2}}\sqrt{r^2-r^2\sin^{2}t}(r\cos(t))dt=r^{2}\int_{0}^{\frac{\pi  }{2}}\cos^2(t)dt

    r^{2}\int_{0}^{\frac{\pi}{2}}\cos^2(t)dt=r^{2}\int  _{0}^{\frac{\pi}{2}}\frac{1}{2}(1+\cos(2t))dt=\fra  c{\pi r^2}{4}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: December 10th 2011, 07:40 PM
  2. Replies: 8
    Last Post: November 27th 2011, 11:18 PM
  3. Replies: 6
    Last Post: July 8th 2010, 06:39 PM
  4. Integrating within a circle?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 24th 2009, 03:45 PM
  5. Replies: 1
    Last Post: May 1st 2008, 08:54 PM

Search Tags


/mathhelpforum @mathhelpforum