A farm pond is (on the surface) in the shape of a perfect circle with a radius of 100 feet. An underwater surveyor maps the bottom of the pond and finds (remarkably!!) that the for any value of r, 0 ≤ r ≤ 100 the depth of the pond at distance r feet from the center of the pond is (150 - 15√r) feet (so the pond is deepest at the center, where r = 0 and decreases as one moves from the center towards the shore.) If the water in the pond has weight density of 64 lb/ft3, find the weight of the water in the pond.
a. Consider a very thin ring of radius rk and centered at the center of the pond. Thus the depth of water at any point in this ring is 150 - 15√rk feet. If the width of the ring is Δrk, write a good estimate for the weight of water beneath this thin ring.
b. Based on your result from part a., write and evaluate an integral for the total weight of water in the pond?