Question

Resolvendo a integral:

Answer 1

Resolvendo a integral pelo Teorema de Fubini (integrais iteradas):

$\displaystyle\int_0^{\pi}\int_0^1{r\cos\theta\,dr\,d\theta}\\\\\\ =\int_0^{\pi}\int_0^1{\cos\theta\cdot r\,dr\,d\theta}\\\\\\ =\int_0^{\pi}{\cos\theta\cdot \left.\left(\dfrac{r^2}{2} \right )\right|_0^1\,d\theta}\\\\\\ =\int_0^{\pi}{\cos\theta\cdot \left(\dfrac{1^2}{2}-\dfrac{0^2}{2} \right )\,d\theta}\\\\\\ =\int_0^{\pi}{\dfrac{1}{2}\,\cos\theta\,d\theta}$

$=\left.\left(\dfrac{1}{2}\,\mathrm{sen\,}\theta\right)\right|_0^{\pi}\\\\\\ =\dfrac{1}{2}\,\mathrm{sen\,}\pi-\dfrac{1}{2}\,\mathrm{sen\,}0\\\\\\ =0.$