Question

encontre a area da regiao sob a curva y=xcosx e o eixo x no intervalo (0,pi/4)

Answer 1

Puesto que $x\cos x\geq 0\;,\; \forall x\in (0,\pi/4)$ entonces

$\displaystyle A=\int _{0}^{\pi/4}x\cos x\,dx\\ \\ A=\int _{0}^{\pi/4}x\,d\sin x\\ \\ A=\left.\left(x\sin x\right)\right|_{0}^{\pi/4}-\int_0^{\pi/4} \sin x dx\\ \\ A= \dfrac{\pi\sqrt{2}}{8}+\left.\left(\cos x\right)\right|_{0}^{\pi/4}\\ \\ \boxed{A= \dfrac{\pi\sqrt{2}}{8}+\dfrac{\sqrt{2}}{2}+1}$