$$\text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. }$$
-D is the rectangular solid whose base is the triangle with vertices at (0, 0), (0, 3), and (3, 3), and whose top lies in the plane z = 5.
A) $\int _ { \pi / 4 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \csc \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$
B) $\int _ { \pi / 4 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \sec \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dzrdr } \mathrm { d } \theta$
C) $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \sec \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$
D) $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \csc \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$

Question

$$\text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. }$$
-D is the rectangular solid whose base is the triangle with vertices at (0, 0), (0, 3), and (3, 3), and whose top lies in the plane z = 5. 
A) $\int _ { \pi / 4 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \csc \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$
B) $\int _ { \pi / 4 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \sec \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dzrdr } \mathrm { d } \theta$
C) $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \sec \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$
D) $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 3 \csc \theta } \int _ { 0 } ^ { 5 } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta$

Quizplus · Accepted Answer

The base of the solid is a right triangle in the first quadrant, with one side along the y-axis from (0,0) to (0,3) and the hypotenuse from (0,3) to (3,3). The limits for $\theta$ are from $\pi/4$ to $\pi/2$ because the line from (0,3) to (3,3) makes a 45-degree angle with the x-axis, and the y-axis corresponds to $\pi/2$. The radius $r$ varies from 0 to the hypotenuse, which can be expressed as $3 \csc \theta$ because $\csc \theta = \frac{1}{\sin \theta}$, and for any point on the hypotenuse, the y-coordinate (which is 3) divided by $\sin \theta$ gives the length of the hypotenuse (or the radius $r$ in polar coordinates). The $z$-coordinate varies from 0 to 5, as given by the plane $z=5$.