Evaluate the improper integral.
-$$\int _ { 0 } ^ { 49 } \int _ { 0 } ^ { 49 } \frac { d y d x } { \sqrt { x } + \sqrt { y } }$$
A) $\frac { 1372 } { 3 } ( 1 - \ln 2 )$
B) $\frac { 2744 } { 3 } ( 1 - \ln 2 )$
C) $\frac { 196 } { 3 } ( 1 - \ln 2 )$
D) $\frac { 392 } { 3 } ( 1 - \ln 2 )$

Question

Quizplus · Accepted Answer

We can use the substitution $u=\sqrt{x}, v=\sqrt{y}$, then $dxdy=dudv$ and the integral becomes
\begin{align*}
\int_0^7\int_0^7\frac{4u v}{u+v}dudv &= 4\int_0^7\int_u^7\frac{u v}{u+v}dudv\
&= 4\int_0^7\int_0^v\frac{u v}{u+v}dudv\
&=4\int_0^7\int_0^v\left(v-\frac{v^2}{u+v}ight) dudv\
&=4\int_0^7\left[v^2u-v^2\ln(u+v)ight]_0^v dv\
&=4\int_0^7\left(\frac{v^3}{3}-v^2\ln(2v)ight)dv\
&= \frac{2744}{3}(1-\ln2)
\end{align*}

Evaluate the Improper Integral ∫049∫049dydxx+y\int _ { 0 } ^ { 49 } \int _ { 0 } ^ { 49 } \frac { d y d x } { \sqrt { x } + \sqrt { y } }∫049​∫049​x​+y​dydx​

Evaluate the Improper Integral $\int _ { 0 } ^ { 49 } \int _ { 0 } ^ { 49 } \frac { d y d x } { \sqrt { x } + \sqrt { y } }$