Evaluate the integral by switching the order of integration.Round your answer to two decimal places.
A)12.39
B)19.36
C)9.73
D)7.73
E)26.57

Question

Evaluate the integral  by switching the order of integration.Round your answer to two decimal places.
A)12.39 
B)19.36 
C)9.73 
D)7.73 
E)26.57

Quizplus · Accepted Answer

We first find the limits of integration for the new order of integration. The region of integration is the rectangle $0\leq y\leq\frac{1}{2}, 0\leq x\leq 2.$ Therefore, we have $$\int_0^2\int_0^{\frac{1}{2}}\frac{4y}{\sqrt{4y^2+x^2}}dydx.$$

Alternatively, we could notice that $\frac{4y}{\sqrt{4y^2+x^2}}$ is the derivative with respect to $y$ of $2\ln(\sqrt{x^2+4y^2}+2y)$, and thus we have $$\int_0^{\frac{1}{2}}\int_0^2\frac{4y}{\sqrt{4y^2+x^2}}dxdy=\int_0^{\frac{1}{2}}\left(2\ln(\sqrt{x^2+4y^2}+2y)\Bigg|_0^2ight)dy.$$ The result is $2\ln(3)-\ln(2)$, which is approximately $1.95$. Therefore, the answer is $\boxed{	extbf{(B) }19.36}$.

Evaluate the Integral by Switching the Order of Integration