12 4+ 75% Problem 4: [20 pts] The following problem is an additional application of the "Slice, Approximate, Integrate" procedure we have discussed in class. Make sure to read the problem carefully. One of the fundamental processes in the human body is the continuous running of blood in the cardiovascular system. The amount of blood that passes through a given cross-section of an artery per unit time is called flux? When the rate u at which blood passes through an area A is constant, the flux @ is given by A - We can assume that the cross-sections of an artery are circles. The rate at which blood flows near the edges is not constant; in fact a good model is where r is the distance from the center of the artery, P is the pressure difference between the ends of the artery, and R are the length and radius, respectively, of the artery, and 7 is the viscosity of the blood. A. [5 pts] Explain why the total flux d through the artery cannot be computed by the formula $ = A . v. a+ 75% Problem 4, continued B. [15 pts] Use the "Slice, Approximate, Integrate" procedure to set up an integral that gives the total flux through the artery, then evalaute the integral. In order to receive full credit, you must clearly explain the following. 1) Why you chose what you did for your variable of integration. 2) What simplifying assumption you use for your approximation step. 3) How to obtain a formula that gives the approximate flux through one of your slices in terms of the variable of integration and its differential. Some possibly helpful hints are on the next page, which you may use in your argument. a 9 W O