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?
Sorry, I really don't understand why dA equals that. My A(y) must be wrong then? What should it be and why?