Change the Cartesian integral to an equivalent polar integral, and then evaluate.
-$\int _ { - 9 } ^ { 9 } \int _ { 0 } ^ { \sqrt { 81 - x ^ { 2 } } } d y d x$
A) $\frac { 81 \pi } { 2 }$
B) $\frac { 729 \pi } { 2 }$
C) $\frac { \pi } { 2 }$
D) $\frac { 9 \pi } { 2 }$

Question

Quizplus · Accepted Answer

The given Cartesian integral represents a semi-circle with radius 9 in the first quadrant. Converting to polar coordinates, the limits for $r$ are from 0 to 9, and for $\theta$, from 0 to $\pi/2$. The polar integral becomes $\int_{0}^{\pi/2} \int_{0}^{9} r dr d\theta$, which evaluates to $\frac{81\pi}{2}$.

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