Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-$\int _ { - 3 } ^ { 3 } \int _ { - \sqrt { 9 - y ^ { 2 } } } ^ { 0 } \frac { \sqrt { x ^ { 2 } + y ^ { 2 } } } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } } } d x d y$
A) $\frac { \pi ( 3 + 2 \ln 4 ) } { 4 }$
B) $\frac { \pi ( 3 + 2 \ln 4 ) } { 2 }$
C) $\frac { \pi ( 3 + \ln 4 ) } { 2 }$
D) $\frac { \pi ( 3 + \ln 4 ) } { 4 }$

Question

Quizplus · Accepted Answer

The given Cartesian integral represents a semicircle in the third quadrant. Converting to polar coordinates, the limits become $ r: 0 $ to $ 3 $ and $ \theta: \pi $ to $ \frac{3\pi}{2} $. The integrand becomes $\frac{r}{1+r}$, and the integral evaluates to $\frac{\pi(3 + 2 \ln 4)}{2}$.

Change the Cartesian Integral to an Equivalent Polar Integral, and Then