Question

Assinale a alternativa que apresenta o valor da integral:

Answer 1

Integral tripla:
$\displaystyle \iiint dx\,dy\,dz=\iint x+c_x\,dy\,dz=\int xy+yc_x+c_y\,dz=\\\\xyz+yzc_x+zc_y+c_z$
Então:
$\displaystyle \iiint\limits_{0~0~0}^{~~~\frac{\pi}{2}~5~2}7r^2dz\,r\,dr\,d\theta$
pode ser resolvida seguindo os passos:

1º) Integrar em relação a z:
$\displaystyle \iiint\limits_{0~0~0}^{~~~\frac{\pi}{2}~5~2}7r^2dz\,r\,dr\,d\theta\\\\i)~~~~\int\limits_{0}^{2}7r^2dz=\left[\frac{}{}7zr^2\right]_{0}^{2}=14r^2-0=\boxed{14r^2}$

2º) Colocar o resultado dentro da integral dupla:
$\displaystyle \iiint\limits_{0~0~0}^{~~~\frac{\pi}{2}~5~2}7r^2dz\,r\,dr\,d\theta=\iint\limits_{0~0}^{~~~\frac{\pi}{2}~5}14r^2r\,drd\theta=\iint\limits_{0~0}^{~~~\frac{\pi}{2}~5}14r^3\,drd\theta$

3º) Integrar em relação a r:
$\displaystyle \iint\limits_{0~0}^{~~~\frac{\pi}{2}~5}14r^3\,drd\theta\\\\i)~~~~\int\limits_{0}^{5}14r^3dr=\left[\frac{14}{4}r^4\right]_{0}^{5}=\frac{14}{4}5^4-0=\frac{8750}{4}=\boxed{\frac{4375}{2}}$

4º) Integrar em relação a theta:
$\displaystyle \iint\limits_{0~0}^{~~~\frac{\pi}{2}~5}14r^3\,drd\theta=\int\limits_{0}^{\frac{\pi}{2}}\frac{4375}{2}d\theta\\\\i)~~~~\int\limits_{0}^{\frac{\pi}{2}}\frac{4375}{2}d\theta=\left[\frac{4375}{2}\theta\right]=\frac{4375\pi}{4}-0=\boxed{\frac{4375\pi}{4}}$

LOGO:
$\displaystyle \iiint\limits_{0~0~0}^{~~~\frac{\pi}{2}~5~2}7r^2dz\,r\,dr\,d\theta=\boxed{\boxed{\frac{4375\pi}{4}}}$

Answer 2

A resposta é:

(pi/4) * 4375

espero ter ajudado pessoal ;)