Provide an Appropriate Response $r = f ( \theta )$ From $\theta = \alpha$

question 66

Short Answer

Provide an appropriate response.
-The area of the surface formed by revolving the graph of $r = f ( \theta )$ from $\theta = \alpha$ to $\theta = \beta$ about the line $\theta = 0$ (the polar axis) is $\mathrm { SA } = \int _ { \alpha } ^ { \beta } 2 \pi \mathrm { r } \sin \theta \sqrt { \mathrm { r } ^ { 2 } + \left( \frac { \mathrm { dr } } { \mathrm { d } \theta } \right) ^ { 2 } } \mathrm {~d} \theta$ . The area of the surface formed by revolving the graph of $\mathrm { r } = \mathrm { f } ( \theta )$ from $\theta = \alpha$ to $\theta = \beta$ about the line $\theta = \frac { \pi } { 2 }$ is $\mathrm { SA } = \int _ { \alpha } ^ { \beta } 2 \pi \mathrm { r } \cos \theta \sqrt { \mathrm { r } ^ { 2 } + \left( \frac { \mathrm { dr } } { \mathrm { d } \theta } \right) ^ { 2 } } \mathrm {~d} \theta$ . Use the idea underlying these two integral formulas to find the surface area of the torus formed by revolving the circle $r =$ a about the line $r = b \sec \theta$ where $0 < a < b$ .

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Definitions:

Freud

An influential figure in psychology, known for his theories of the unconscious mind, the mechanism of repression, and the psychoanalytic model of the mind.

Pleasure Seeking

The pursuit of activities and experiences that are expected to bring joy or satisfaction.

Id's Drives

Basic, instinctual forces or desires that seek immediate gratification, according to Freudian psychoanalytic theory.

External World

The objective reality or the physical world that exists independently of our thoughts or perceptions.