Use the Shell Method to Set Up and Evaluate the Integral

question 27

Multiple Choice

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. $y = x ^ { 6 } , x = 0 , y = 64$

• A) \(V = 2 \pi \int _ { 0 } ^ { 64 } y \left( y ^ { \frac { 1 } { 6 } } \right) d y = \frac { 98,304 } { 13 } \pi\)
• B) \(V = 2 \pi \int _ { 0 } ^ { 64 } y \left( y ^ { - \frac { 1 } { 6 } } \right) d y = \frac { 49,152 } { 13 } \pi\)
• C) \(V = 2 \pi \int _ { 0 } ^ { 64 } y \left( y ^ { \frac { 1 } { 6 } } \right) d y = \frac { 24,576 } { 13 } \pi\)
• D) \(V = 2 \pi \int _ { 0 } ^ { 64 } y \left( y ^ { \frac { 1 } { 6 } } \right) d y = \frac { 48,384 } { 13 } \pi\)
• E)\(V = 2 \pi \int _ { 0 } ^ { 64 } y \left( y ^ { \frac { 1 } { 6 } } \right) d y = \frac { 768 } { 13 } \pi\)

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Definitions:

Significant Correlation

A relationship between variables that is strong enough to not be attributed to chance, according to statistical tests.

Correlation Coefficient

A numerical index that reflects the degree of relationship between two variables ranging from -1 (perfect negative correlation) to +1 (perfect positive correlation).

Research Question

A clear, focused, concise, and answerable question that guides a study or research project.

Significantly Correlated

A statistical relationship between two variables where changes in one variable are reliably associated with changes in another variable, often verified through statistical tests.