Evaluate the following integral by first converting to an integral in polar coordinates.
∫
0
−
2
∫
√
4
−
y
2
−
√
4
−
y
2
x
2
d
x
d
y
∫−20∫−4−y24−y2x2dxdy
1. Use Lagrange Multiplier to determine the dimensions of a rectangular box, open at the top, having a volume of 32 cubic feet and requiring the least amount of material for its construction.
two boat each move a velocity v relative to the water and both cross a river of breadth a running with uniform velocity V.they start together one boat crossing by shortest path and the otherin shortest time.find the difference between the time of arrival.