Question

Calcule as integrais duplas em regiões não-retangulares;
a)∫_0^2▒∫_(x^2)^x▒〖y^2 dydx;〗
b)∫_0^π▒∫_0^(cos y)▒〖x sen y dxdy; 〗

Answer 1

a) $\displaystyle\int\limits_{0}^{2}\int\limits_{x^{2}}^{x}{y^{2}\,dy\,dx}$

$=\displaystyle\int\limits_{0}^{2}{\left.\dfrac{y^{3}}{3}\right|_{x^{2}}^{x}\,dx}\\ \\ \\ =\int\limits_{0}^{2}{\left(\dfrac{x^{3}}{3}-\dfrac{(x^{2})^{3}}{3} \right )\,dx}\\ \\ \\ =\int\limits_{0}^{2}{\left(\dfrac{x^{3}}{3}-\dfrac{x^{6}}{3} \right )\,dx}\\ \\ \\ =\left.\left(\dfrac{x^{4}}{12}-\dfrac{x^{7}}{21} \right )\right|_{0}^{2}\\ \\ \\ =\dfrac{2^{4}}{12}-\dfrac{2^{7}}{21}\\ \\ \\ =\dfrac{16}{12}-\dfrac{128}{21}\\ \\ \\ =\dfrac{112-512}{84}\\ \\ \\ =-\dfrac{400}{84}\\ \\ \\ =-\dfrac{100}{21}$


b) $=\displaystyle\int\limits_{0}^{\pi}\int\limits_{0}^{\;\cos(y)}{x\,\mathrm{sen}(y)\,dx\,dy}$

$=\displaystyle\int\limits_{0}^{\pi}{\mathrm{sen}(y)\cdot \left.\left(\dfrac{x^{2}}{2} \right )\right|_{0}^{\cos(y)}\,dy}\\ \\ \\ =\int\limits_{0}^{\pi}{\mathrm{sen}(y)\cdot \dfrac{\cos^{2}(y)}{2}\,dy}\\ \\ \\ =-\dfrac{1}{2}\int\limits_{0}^{\pi}{\cos^{2}(y)\cdot [-\mathrm{sen}(y)]\,dy}\\ \\ \\ =-\dfrac{1}{2}\cdot \left.\left(\dfrac{\cos^{3}(y)}{3} \right )\right|_{0}^{\pi}\\ \\ \\ =-\dfrac{1}{2}\cdot \left(\dfrac{\cos^{3}(\pi)}{3}-\dfrac{\cos^{3}(0)}{3} \right )\\ \\ \\ =-\dfrac{1}{2}\cdot \left(\dfrac{(-1)^{3}}{3}-\dfrac{1^{3}}{3} \right )\\ \\ \\ =-\dfrac{1}{2}\cdot \left(-\dfrac{1}{3}-\dfrac{1}{3} \right )\\ \\ \\ =-\dfrac{1}{\diagup\!\!\!\! 2}\cdot \left(-\dfrac{\diagup\!\!\!\! 2}{3} \right )\\ \\ \\ =\dfrac{1}{3}$