1. A lamina occupies the part of the disk x 2 + y 2 ? 36 in the first quadrant.

Question

1. A lamina occupies the part of the diskx2+y2?36in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.

What is (x,y)?

2. The boundary of a lamina consists of the semicircles

and

3. A lamina occupies the region inside the circlex2+y2=6ybut outside the circlex2+y2=9.Find the center of mass if the density at any point is inversely proportional to its distance from the origin. What is(x,y)?

4.An agricultural sprinkler distributes water in a circular pattern of radius120ft. It supplies water to a depth ofe?rfeet per hour at a distance ofrfeet from the sprinkler.

(a) If0 <R?120,what is the total amount of water supplied per hour to the region inside the circle of radiusRcentered at the sprinkler? ________ft^3

b) Determine an expression for the average amount of water per hour per square foot supplied to the region inside the circle of radiusR.
_______ft3(per hour per square foot)

5. Evaluate the iterated integral by converting to polar coordinates.

6. Use polar coordinates to find the volume of the given solid.

Explanation / Answer

Can you integrate? Here are the definition you need to know: The total mass is just the (infinite) sum of the (infinitesimal) masses of the infinitesimal areas (using: mass = density

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