Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R.
$\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A$
$R$ : triangle bounded by $y = 2 x , y = 6 x$, and $x = 5$
A) $5 ( \log ( 37 ) - \log ( 5 ) )$
B) $\frac { 2 } { 5 } ( \log ( 37 ) - \log ( 5 ) )$
C) $\frac { 5 } { 2 } ( \log ( 37 ) - \log ( 5 ) )$
D) $\frac { 5 } { 2 } ( \log ( 5 ) - \log ( 37 ) )$
E) $\frac { 1 } { 3 } ( \log ( 37 ) - \log ( 5 ) )$

Question

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. 
$\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A$
$R$ : triangle bounded by $y = 2 x , y = 6 x$, and $x = 5$
A) $5 ( \log ( 37 ) - \log ( 5 ) )$
B) $\frac { 2 } { 5 } ( \log ( 37 ) - \log ( 5 ) )$
C) $\frac { 5 } { 2 } ( \log ( 37 ) - \log ( 5 ) )$
D) $\frac { 5 } { 2 } ( \log ( 5 ) - \log ( 37 ) )$
E) $\frac { 1 } { 3 } ( \log ( 37 ) - \log ( 5 ) )$

Quizplus · Accepted Answer

The more convenient order of integration is dydx, as the function inside the integral is more easily handled when integrating with respect to y first. The limits for y are from $2x$ to $6x$, and for x from 0 to 5. The integral becomes:$$\int_{0}^{5} \int_{2x}^{6x} \frac{y}{x^2+y^2} dy dx$$Solving this integral, we find that the answer is $\frac{5}{2}(\log(37) - \log(5))$, which corresponds to choice C. This is because the integral simplifies to a logarithmic function of the upper and lower bounds of x after integrating with respect to y and then x, showcasing the logarithmic relationship between the bounds of integration and the final answer.

Set Up an Integral for Both Orders of Integration, and Use