(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Derive via integration the first moment of areaQof a semicircle with radiusr.

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?

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# Deriving the first moment of area of semicircle

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