Set Up a Riemann Sum Approximating the Volume of the Torus

question 75

Multiple Choice

Set up a Riemann sum approximating the volume of the torus (donut) obtained by rotating the circle $(x-6) ^{2}+y^{2}=1$ about the y-axis.

• A) \(\sum \pi\left(6+\sqrt{1-y^{2}}\right)^{2} \Delta y\)
• B) \(\sum \pi\left(6-\sqrt{1-y^{2}}\right)^{2} \Delta y\)
• C) \(\sum \pi\left(\left(6+\sqrt{1-y^{2}}\right)^{2}+\left(6-\sqrt{1-y^{2}}\right)^{2}\right) \Delta y\)
• D) \(\sum \pi\left(\left(6+\sqrt{1-y^{2}}\right)^{2}-\left(6-\sqrt{1-y^{2}}\right)^{2}\right) \Delta y\)

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Definitions:

Trade

The exchange of goods and services between people or entities.

Total Surplus

The sum of consumer surplus and producer surplus in a market, reflecting the total net benefit to society from the production and consumption of goods and services.

Trade

The exchange of goods and services between parties, which can be within or across national borders.

Consumer Surplus

The difference between the total amount that consumers are willing and able to pay for a good or service and the total amount they actually do pay.