If the velocity distribution of a fluid flowing through a pipe is known (Figure), the flow rate Q (that is, the volume of water passing through the pipe per unit time) can be computed by Q = ∫υdA, where υ is the velocity and A is the pipe’s cross-sectional area. (To grasp the meaning of this relationship physically, recall the close connection between summation and integration.) For a circular pipe. A = πr² and dA = 2πr dr. Therefore,

where r is the radial distance measured outward from the center of the pipe. If the velocity distribution is given by

where r0 is the total radius (in this case, 3 cm), compute Q using the multiple-application trapezoidal rule. Discuss theresults.

(2rr)dr