$$\text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. }$$

question 427

Multiple Choice

$$\text { Set up the iterated integral for evaluating } \iint _ { D } \int f ( r , \theta , z ) d z r d r d \theta \text { over the given region D. }$$
- $\mathrm { D }$ is the solid right cylinder whose base is the region between the circles $r = 3 \sin \theta$ and $r = 4 \sin \theta$ , and whose top lies in the plane $z = 5 - x - y$ .

• A) \(\int _ { 0 } ^ { 2 \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta + \sin \theta ) } f ( r , \theta , z ) d z r d r d \theta\)
• B) \(\int _ { 0 } ^ { \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta - \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta\)
• C) \(\int _ { 0 } ^ { 2 \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta - \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta\)
• D) \(\int _ { 0 } ^ { \pi } \int _ { 3 \sin \theta } ^ { 4 \sin \theta } \int _ { 0 } ^ { 5 - r ( \cos \theta + \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta\)

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