If R is the region in the first quadrant bounded by $x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9$ then $\iint$_R $e ^ { x ^ { 2 } + y ^ { 2 } } d A$ in polar form is A) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } e ^ { r 2 } d r \right] d \theta$ B) $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$ C) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$ D) $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 2 } ^ { 3 } e ^ { r 2 } d r \right] d \theta$ E) $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 2 } ^ { 3 } r e ^ { r 2 } d r \right] d \theta$

Question

Quizplus · Accepted Answer

The correct integral in polar coordinates includes the Jacobian determinant $r$ when changing from Cartesian to polar coordinates, and the region is in the first quadrant between circles of radius 1 and 3, so the limits for $r$ are incorrect in all options except for the limits of $\theta$, which correctly span from 0 to $\frac{\pi}{2}$ for the first quadrant.

If R Is the Region in the First Quadrant Bounded x2+y2=1,x2+y2=9x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9x2+y2=1,x2+y2=9

If R Is the Region in the First Quadrant Bounded $x ^ { 2 } + y ^ { 2 } = 1 , x ^ { 2 } + y ^ { 2 } = 9$