For each system below, determine whether it displays compensatory growth, depensatory growth, or critical depensation. Justify your answer in each case. (b) N = rN²e¯, where r > 0, K > 0.

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8 For a sphere of radius r = a, find by integration (a) its surface area, (b) the centroid of the curved surface of a hemisphere, (c) the moment of inertia of the whole spherical shell about a diameter assuming constant area density, (d) the volume of the ball r≤a, (e) the centroid of a solid half ball.

7 (a) Find the moment of inertia of a circular disk of uniform density about an axis through its center and perpendicular to the plane of the disk. (b) Find the moment of inertia of a solid circular cylinder of uniform density about its central axis. (c) theorem. Do (a) by first calculating the moment of inertia about a diameter and then using the perpendicular axis

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3. Consider the following theorem: Theorem: If n is an odd integer, then n³ is an odd integer. Note: There is an implicit universal quantifier for this theorem. Technically we could write: For all integers n, if n is an odd integer, then n³ is an odd integer. (a) Explore the statement by constructing at least three examples that satisfy the hypothesis, one of which uses a negative value. Verify the conclusion is true for each example. You do not need to write your examples formally, but your work should be easy to follow. (b) Pick one of your examples from part (a) and complete the following sentence frame: One example that verifies the theorem is when n = We see the hypothesis is true because and the conclusion is true because (c) Use the definition of odd to construct a know-show table that outlines the proof of the theorem. You do not need to write a proof at this time.

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Please ensure that all parts of the question are answered thoroughly and clearly. Include a diagram to help explain answers. Make sure the explanation is easy to follow. Would appreciate work done written on paper. Thank you.

Explore this statement by constructing at least three examples, one of which must be a negative integer. Indicate if the statement is true or false for each example.