Evaluate the iterated integral by switching the order of integration.Round your answer to three decimal places.
A)0.538
B)0.558
C)30.538
D)1.538
E)4.538

Question

Evaluate the iterated integral  by switching the order of integration.Round your answer to three decimal places.
A)0.538 
B)0.558 
C)30.538 
D)1.538 
E)4.538

Quizplus · Accepted Answer

We can switch the order of integration by integrating with respect to $x$ first and then with respect to $y$:
$$\int_0^{\pi/6} \int_0^{1/\cos(x)} (x+y^2) \, dy \, dx$$
Now we can evaluate the integral step by step:
\begin{align*}
\int_0^{\pi/6} \int_0^{1/\cos(x)} (x+y^2) \, dy \, dx &= \int_0^{\pi/6} \left[ xy + \frac{1}{3}y^3 ight]_0^{1/\cos(x)} \, dx \
&= \int_0^{\pi/6} \left( x \cdot \frac{1}{\cos(x)} + \frac{1}{3\cos^3(x)} ight) \, dx \
&= \left[ \sin(x) + \frac{1}{2\cos^2(x)} ight]_0^{\pi/6} \
&= \left( \frac{1}{2} + \frac{2\sqrt{3}}{9} ight) - \frac{3}{2} \
&= \frac{2\sqrt{3}}{9} - 1 \
&\approx 0.538
\end{align*} Therefore, the answer is (A).

Evaluate the Iterated Integral by Switching the Order of Integration.Round