Math 152 Engineering Mathematics IIHonors Sections 201–202Second ExaminationFall 2004Work all five problems. These are essay questions. To obtain maximalcredit, show your work and explain your reasoning.1. Atmospheric pressure varies with the weather and with the positionon the globe. For reference purposes, the international standard atmo-sphere is defined to be 101, 325 Pa. (Recall that 1 Pa = 1 N/m2.)The mathematician Evangelista Torricelli (shown here on an Italianpostage stamp) conceived the idea of the barometer in 1643 and foundby experiment that atmospheric pressure is equal to the pressure at thebottom of a column of mercury 0.76 m high.Determine the value of the ratio(density of mercury)(density of water).2. You know how to compute the arc length of a curve in the plane byintegration. This problem asks you to generalize to the case of a curvein three-dimensional space by using the arc length elementds =p(dx)2+ (dy)2+ (dz)2.Find the length of the helix that is given in parametric form by theequationsx(t) = cos(t)y(t) = sin(t)z(t) = tfor 0 ≤ t ≤ 6π.3. Find a function f with the property that for every positive number a,a(¯x, ¯y)¯y = ¯x + 1y = f(x)the centroid of the region in the first quadrant under the graph of fbetween 0 and a has a y-coordinate that is exactly one unit bigger thanthe x-coordinate.November 3, 2004 Page 1 of 2 Dr. BoasMath 152 Engineering Mathematics IIHonors Sections 201–202Second ExaminationFall 20044. Solve the differential equationdydx=xy+yx.Hint: Introduce a new dependent variable u via u = y/x. You shouldbe able to transform the given equation into a separable differentialequation (in the variables u and x) that you know how to solve.5. You know that the harmonic series∞Xn=11ndiverges. This problem quan-tifies the rate of growth of the partial sums of the harmonic series.By suitably interpreting the figure below, show that the limitlimn→∞µ1 +12+13+ · · · +1n− log(n + 1)¶(a) exists, and(b) has a value that is between 0 and 1.1 2 3 4y = 1/x(The value of the limit is known as Euler’s constant, usually denotedby the Greek letter γ. The value of γ is approximately 0.577.)November 3, 2004 Page 2 of 2 Dr.
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