Question

o valor da integral dupla de 0 ate $\pi$/2 e 0 ate 2 r^4cos(2∅) drd∅ é:

a) -pi/2
b) -2
c) 2
d) 0
e) pi/2

Answer 1

Olá


Alternativa correta, letra D) 0



$\displaystyle \mathsf{ \int\limits^ \frac{\pi}{2} _0 \int\limits^2_0 {r^4\cdot cos(2\theta)} \, dr d\theta }\\\\\\\mathsf{ \int\limits^ \frac{\pi}{2} _0 \left[\int\limits^2_0 {r^4\cdot cos(2\theta)} \, dr \right] d\theta}\\\\\\\\\mathsf{ \int\limits^ \frac{\pi}{2} _0 \left[ \frac{r^5}{5} \cdot cos(2\theta)\right]^{2}_0d\theta}\\\\\\\\\mathsf{ \int\limits^ \frac{\pi}{2} _0 \left[ \frac{2^5}{5} \cdot cos(2\theta)~-~0\right]d\theta}$

$\displaystyle\mathsf{ \int\limits^ \frac{\pi}{2} _0 \frac{32}{5}cos(2\theta) d\theta}\\\\\\\mathsf{ \frac{32}{5} \int\limits^ \frac{\pi}{2} _0 cos(2\theta) d\theta\qquad\qquad\qquad\qquad \boxed{\int cos(\alpha x)~=~ \frac{sen(\alpha x)}{\alpha} }}\\\\\\\mathsf{ \frac{32}{5}\left( \frac{sen(2\theta)}{2} \right)\bigg|^ \frac{\pi}{2}_0 }\\\\\\\mathsf{\left( \frac{32\cdot sen(2\theta)}{10} \right)\bigg|^ \frac{\pi}{2}_0 }$

$\displaystyle \mathsf{\left( \frac{32sen(2\cdot \frac{\pi}{2}) }{10} \right)~-~ \mathsf{\left( \frac{32sen(2\cdot 0) }{10} \right)}}\\\\\\\mathsf{ \underbrace{\frac{32\cdot sen(\pi)}{10}}_{=0}~-~ \underbrace{\frac{32sen(0)}{10} }_{=0}}}\\\\\\\mathsf{\boxed{=0}}$