Question

Sea p > 0. calcular la masa de una lámina delimitada por x = p y la parábola y 2 = 2px, si la función de densidad de masa en cada punto es ρ(x, y) = pxy2 , .

Answer 1

$\displaystyle M=\iint\limits_{D}\rho(x,y)~dx~dy\\ \\ \\ M=\iint\limits_{D}pxy^2~dx~dy\\ \\ \\ M=p\iint\limits_{D}xy^2~dx~dy\\ \\ \\ \text{Donde: }D=\left\{(x,y):0\leq x\leq 2p, 0\leq y\leq \sqrt{2px}\right\}\\ \\ \\ M=p\int\limits_{0}^{2p}\int\limits_{0}^{\sqrt{2px}}xy^2~dy~dx\\ \\ \\ M=p\int\limits_{0}^{2p}x\left.\left(\dfrac{y^3}{3}\right)\right|_{0}^{\sqrt{2px}}dx\\ \\ \\ M=\dfrac{p}{3}\int\limits_{0}^{2p}x\sqrt{2px}^3dx$


$\displaystyle M=\dfrac{p\sqrt{2p}^3}{3}\int\limits_{0}^{2p}x\sqrt{x}^3dx\\ \\ \\ M=\dfrac{p\sqrt{2p}^3}{3}\int\limits_{0}^{2p}x^{5/2}dx\\ \\ \\ M=\dfrac{p\sqrt{2p}^3}{3}\left.\left(\dfrac{2}{7}x^{7/2}\right)\right|_{0}^{2p}\\ \\ \\ M=\dfrac{2p\sqrt{2p}^3}{21}\cdot (2p)^{7/2}\\ \\ \\ \boxed{M=\dfrac{64p^6}{21}}$