Math  Calculus
posted by Elsi .
Water is flowing in a cylindrical pipe of radius 1 inch. Because water is viscous and sticks to the pipe, the rate of flow varies with distance from the center. The speed of the water at distance r inches from the center is 10(1(r^2)) per second. What is the rate (in cubic inches per second) at which water is flowing through the pipe?

consider the amount of water that flows through a crosssection of the pipe in 1 second. It is a conical solid, high in the center and zero at the edges.
At radius r, the volume of the cylindrical shell of water is 2pi*r*h dr, where h = 10(1r^2)
So, the amount across the whole crosssection is
integral[0,1] 2pi*r*10(1r^2) dr
= 10pi r^2  5pi r^4 [0,1]
= 5pi in^3
since that's the volume in 1 second, the rate is 5pi in^3/s
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