Deriving the first moment of area of semicircle

  1. 1. The problem statement, all variables and given/known data
    Derive via integration the first moment of area Q of a semicircle with radius r.

    2. Relevant equations
    [tex]Q = \int_{A} y dA[/tex]

    [tex] A_{semicircle} = \frac{\pi r^{2} }{2}[/tex]

    3. The attempt at a solution
    [tex] A = \frac{\pi r^{2} }{2}[/tex]
    [tex] A(y) = \frac{\pi y^{2} }{2}[/tex]
    [tex] dA = \pi y dy[/tex]

    [tex]Q = \int^{y=r}_{y=0} y dA[/tex]
    [tex] = \int^{r}_{0} \pi y^{2} dy[/tex]
    [tex] = \frac{\pi}{3} [y^{3}]^{r}_{0}[/tex]

    [tex] Q = \frac{\pi r^{3}}{3}[/tex]

    But the answer is [tex]\frac{2 r^{3} }{3}[/tex], which my textbook derived from the equation [tex]Q = (area) \times (centroidal height) [/tex]. I want to know how to derive the Q for any shape without knowing its centroidal height beforehand. Can someone help me out with why I got a different and wrong answer?
  2. jcsd
  3. nvn

    nvn 2,124
    Science Advisor
    Homework Helper

    Elbobo: dA is not pi*y*dy. Hint: Shouldn't dA instead be, dA = 2[(r^2 - y^2)^0.5]*dy? Try again.
  4. Sorry, I really don't understand why dA equals that. My A(y) must be wrong then? What should it be and why?
  5. nvn

    nvn 2,124
    Science Advisor
    Homework Helper

    Elbobo: A(y) = integral(dA), integrated from y = y1 to y = r. In your particular case, y1 = 0.
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