Solve the Problem $\mathrm { D }$ Be the Region Bounded Below by the Cone

question 106

Multiple Choice

Solve the problem.
-Let $\mathrm { D }$ be the region bounded below by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and above by the sphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ . Set up the triple integral in cylindrical coordinates that gives the volume of $D$ using the order of integration $\mathrm { dr } \mathrm { dz } \mathrm { d } \theta$ .

• A) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 / \sqrt { 2 } } \int _ { 0 } ^ { z } r d r d z d \theta + \int _ { 0 } ^ { 2 \pi } \int _ { 2 / \sqrt { 2 } } ^ { 2 } \int _ { 0 } ^ { \sqrt { 8 - z ^ { 2 } } } r d r d z d \theta\)
• B) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 / 2 } \int _ { 0 } ^ { z } r d r d z d \theta + \int _ { 0 } ^ { 2 \pi } \int _ { 2 / 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 8 - z ^ { 2 } } } r d r d z d \theta\)
• C) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 / 2 } \int _ { 0 } ^ { \mathrm { z } } \mathrm { rdrdz } \mathrm { d } \theta + \int _ { 0 } ^ { 2 \pi } \int _ { 2 / 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - \mathrm { z } ^ { 2 } } } \mathrm { rdr } \mathrm { dz } \mathrm { d } \theta\)
• D) \(\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 / \sqrt { 2 } } \int _ { 0 } ^ { z } r d r d z d \theta + \int _ { 0 } ^ { 2 \pi } \int _ { 2 / \sqrt { 2 } } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - z ^ { 2 } } } r d r d z d \theta\)

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Definitions:

Pure Monopolist

A market structure where a single firm completely dominates the market, with no close substitutes for its product, allowing it to control prices.

Discriminating

The act of making a distinction or differentiating between items, people, or situations, often based on specific criteria.

Price Discrimination

A pricing strategy where a seller charges different prices for the same product or service to different consumers, based on their willingness to pay.

Durable Good

A consumer good with an expected life (use) of three or more years.