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

question 246

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 right circular cylinder whose base is the circle $\mathrm { r } = 6 \cos \theta$ in the $x y - p l a n e$ and whose top lies in the plane $z = 8 - x - y$ .

• A) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 \sin \theta } \int _ { 0 } ^ { 8 - \sin \theta - \cos \theta } f ( r , \theta , z ) d z r d r d \theta\)
• B) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 6 \cos \theta } \int _ { 0 } ^ { 8 - r ( \cos \theta + \sin \theta ) } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta\)
• C) \(\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 6 \cos \theta } \int _ { 0 } ^ { 8 - r ( \cos \theta + \sin \theta ) } f ( r , \theta , z ) d z r d r d \theta\)
• D) \(\int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 6 \sin \theta } \int _ { 0 } ^ { 8 - \sin \theta - \cos \theta } \mathrm { f } ( \mathrm { r } , \theta , \mathrm { z } ) \mathrm { dz } \mathrm { rdr } \mathrm { d } \theta\)

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Definitions:

Interval Estimator

An interval estimator provides a range of values, defined by an upper and lower bound, that is likely to contain the population parameter of interest.

Sample Mean

The average of a subset of numbers from a larger set, representing the central tendency of that subset.

Consistent Estimator

An estimator that, as the sample size increases, converges in probability to the value of the parameter being estimated.

Population Mean

The average of all the values in a population.