Question

o valor da integral iterada ∫π/20∫cosθ0∫r20rsinθdzdrdθ é igual a:

Escolha uma:
a. 0,2
b. 0,05
c. π
d. 1
e. π/2

Answer 1

$\int\limits^\frac\pi2_0 {\int\limits^{\cos\theta}_0 {\int\limits^{r^2}_0 {r\sin\theta} \, dz } \, dr } \, d\theta =\\\\\\\int\limits^\frac\pi2_0 {\int\limits^{\cos\theta}_0 {[rz\sin\theta]^{r^2}_0} \, dr } \, d\theta= \\\\\\\int\limits^\frac\pi2_0 {\int\limits^{\cos\theta}_0 {r^3\sin\theta} \, dr } \, d\theta =\\\\\\\int\limits^\frac\pi2_0 {\left[\dfrac{r^4\sin\theta}{4}\right]^{\cos\theta}_0} } \, d\theta=\\\\\\\int\limits^\frac\pi2_0 {\left[\dfrac{\cos^4\theta\sin\theta}{4}\right] } \, d\theta$

Agora temos uma integral comum:

$\int\limits^\frac\pi2_0 {\left[\dfrac{\cos^4\theta\sin\theta}{4}\right] } \, d\theta\\\\\\\dfrac14\int\limits^\frac\pi2_0 {\cos^4\theta\sin\theta} \, d\theta=\\\\\\u=\cos\theta\\\\du=-\sin\theta \ d\theta\\\\\\\dfrac14\int\limits^\frac\pi2_0 {-u^4} \, du=\\\\\\\dfrac14\left[\dfrac{-u^5}{5}\right]^\frac\pi2_0=\\\\\\\dfrac14\left[-\dfrac{\cos^5\theta}{5}\right]^\frac\pi2_0=\dfrac14\left[\dfrac15\right]=\dfrac{1}{20}=\boxed{0.05}$