Use the Method of Cylindrical Shells to Find the Volume $$y = \frac { 1 } { x \left( x ^ { 2 } + 144 \right) } , \quad y = 0 , x = 1 , x = 12 \text {; the } y \text {-axis }$$

question 45

Short Answer

Use the method of cylindrical shells to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated axis.Sketch the region and a representative rectangle. $$y = \frac { 1 } { x \left( x ^ { 2 } + 144 \right) } , \quad y = 0 , x = 1 , x = 12 \text {; the } y \text {-axis }$$

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Definitions:

Corollary

A proposition that follows with little or no proof required from one already proven, serving as a natural consequence or result.

Deep Ecology

A philosophy that promotes the inherent worth of living beings regardless of their instrumental utility to human needs, deeply questioning mainstream environmentalism's emphasis on resource management.

Inherent Value

The intrinsic worth of something, not based on its utility to humans, often referenced in environmental ethics and conservation.

Material Wealth

The abundance of physical, tangible assets and resources that an individual or society possesses, often considered an indicator of economic well-being.