Deriving the first moment of area of semicircle

by Elbobo
Tags: deriving, moment, semicircle
 P: 145 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 $$Q = \int_{A} y dA$$ $$A_{semicircle} = \frac{\pi r^{2} }{2}$$ 3. The attempt at a solution $$A = \frac{\pi r^{2} }{2}$$ $$A(y) = \frac{\pi y^{2} }{2}$$ $$dA = \pi y dy$$ $$Q = \int^{y=r}_{y=0} y dA$$ $$= \int^{r}_{0} \pi y^{2} dy$$ $$= \frac{\pi}{3} [y^{3}]^{r}_{0}$$ $$Q = \frac{\pi r^{3}}{3}$$ But the answer is $$\frac{2 r^{3} }{3}$$, which my textbook derived from the equation $$Q = (area) \times (centroidal height)$$. 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?