An Object Is in the Shape Drawn Below; Its Boundary $y=2 x^{2}$

question 68

Multiple Choice

An object is in the shape drawn below; its boundary is obtained by rotating the parabola $y=2 x^{2}$ (for 0 $\le$ x $\le$ 1) around the y axis.(Units are in centimeters.) Suppose that the density of this object varies with height according to the rule $\rho$ (y) = 12 (2 - y) grams/cm³.Which of the following Riemann sums computes (approximately) the mass in grams of this object?

• A) \(\sum_{y=0}^{2} \pi \cdot y^{2} \cdot 12(2-y) \Delta y\)
• B) \(\sum_{y=0}^{2} \pi \cdot \frac{y^{2}}{4} \cdot 12(2-y) \Delta y\)
• C) \(\sum_{y=0}^{2} \pi \cdot \frac{y}{2} \cdot 12(2-y) \Delta y\)
• D) \(\sum_{y=0}^{2} \pi \cdot y \cdot 12(2-y) \Delta y\)

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Definitions:

Creativity

The ability to produce original and valuable ideas or things, involving imagination and the willingness to explore.

Meditation

A practice of concentrated focus upon a sound, object, visualization, the breath, movement, or attention itself to increase awareness of the present moment.

Respiration

Respiration is the process by which living organisms take in oxygen and release carbon dioxide, involving the exchange of gases as a part of metabolism.

Zen Meditation

A form of meditation originating in Buddhism that emphasizes sitting in reflective mindfulness to gain insight into the nature of existence and personal enlightenment.