Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, are from the mathematicalfolklore, or are independently marked.Here’s a hint: drawing pictures — e.g. sketching graphs of functions — will always make theproblem easier.Even More Applications of IntegrationThe velocity v of a liquid flowing at distance r from the center of a cylindrical channel of radiusR is given by v(r) =14Pl1η(R2− r2), where η is the viscosity of the liquid, P is the pressure, and lis the length of the channel. (The fraction P/l is the pressure per unit length. It’s not surprisingthat this quantity determines the velocity; what’s surprising is the sensitivity to the radius.) Theflux through a channel is the amount of fluid across a cross-section of the channel per unit time.We integrateRRr=0v dA, where dA = 2πr dr is the infinitesimal area at radius r, to conclude thatthe flux in a cylindrical channel is φ =π8Pl1ηR4. [We use φ for “flux”, so that we can save the letterF for “force”.]Another extremely important use of integration is in the definition of work. If an object movesa distance x against a force F , the work done on the object is W = F x. If the force varies withlocation, so that F = F (x), then the work is W =RbaF (x)dx, where the object moves from locationa to location b.1. (a) Prove the following theorem of Archimedes: an object fully submerged in a fluid expe-riences an upward “buoyant” force equal to the weight of the fluid that would fill thevolume of the object. Hint: consider first a normally-oriented rectangular box (you canprove the theorem for boxes without calculus). Then use integral-style arguments toprove the theorem for arbitrary objects.(b) What is the work required to push an object with volume V a distance h down underwater?2. In a spring with spring constant k, the force at location x is −kx. Find the work required tomove an object from location a to location b.3. In a pendulum with mass m and length l, the force required to move the bob up along thearc of the pendulum, when it has already moved a distance x from the bottom of the arc, isgiven by F (x) =12πmg sinx2πl. Find the work required to move the mass a distance x alongthe arc of the pendulum, if it starts at the bottom of the arc.4. (a) A charged particle at a distance x from another charge feels a force F (x) = k/x2, where kdepends on the two charges. Find the work required to move from distance a to distanceb. Find the work required to move the charge to distance a from ∞.(b) If the electric field is created not be another charge but by an electric dipole, then theforce is F (x) = k/x3for some k. Find the work required to move from distance a todistance b. Find the work required to move the charge to distance a from ∞.5. Lets say an object moves along the curve y = f(x) for x ∈ [a, b], against a frictive force F (x).Then the total work performed isRbaF (x) ds, where ds =pdx2+ dy2=p1 + (f0(x))2dx.1(a) If the friction is F (x) = x and the curve is y = x2, x ∈ [0, 1], what is the total work?(b) If the force of friction is constant, what is the relationship between work and arclength?6. Rather than a circular pipe, let’s consider a channel that consists of two horizontal planesseparated at distance H — really we’re considering a pipe that consists of a very wide rectanglewith height H, and width much much more than H. We consider a liquid flowing throughthe pipe under laminar conditions. Let h measure the height from the top of the channel;then the velocity v(h) of the liquid at height h satisfies the differential equationd2vdh2=1ηPlwhere P is the pressure on the channel and l is the length of the channel, so that P/l is thepressure drop per unit length.(a) We haven’t talked about differential equations yet. Nevertheless, we will solve this one.Assume that P is constant throughout the channel, and find the most general functionv(h) that satisfies the above differential equation.(b) The laws of physics require that the velocity of a fluid is zero at the edge of a channel:v(0) = 0 = v(H). Given these conditions and your answer to part (a), find an explicitformula for v(h).(c) Find the linear flux across across the channel by integratingRH0v(h) dh. The actual fluxis this number times the width of the channel.(d) Let’s now suppose that the force of gravity is quite strong, so that the pressure P dependson the height h, via P (h) = ρgh + P0, where ρ is the density of the liquid and g is theacceleration due to gravity (P0is the pressure applied by the pump). Repeat steps (a)through (c) above with this pressure.(e) Finally, let’s suppose that the “liquid” consists of electrons flowing in a conductor, andthat “gravity” is actually an external vertical electric field. Then P (h) = K/(h+A)+P0for some some numbers K, A, and P0(A is essentially the distance from the channel tothe external field, and K depends on the strength of the field). Repeat steps (a) through(c) for this pressure.7. Recall that when the curve y = f(x) is rotated around the x-axis, the surface area of thepiece of curve corresponding to the interval [x, x + dx] is dA = 2πf (x) ds, where ds =p1 + (f0(x))2dx. Find the total hydrostatic force felt by a sphere of radius 1 m submergedunder water so that the center is at a depth h (with h ≥ 1 m). Hint: orient the axes with xpointing down through the center of the sphere and x = 0 corresponding to the surface of thewater.8. Not a physics problem. § Pareto’s Law of Income states that the number of peoplewith incomes between x = a and x = b is N =RbaAx−kdx for some constants A and k.It is remarkably accurate for large incomes. The average income of these people is ¯x =1NRbaAx1−kdx.(a) There are only finitely many people in the world. For Pareto’s Law to be true for largeincomes, what conditions must we impose on k?(b) There is only finitely much money in the world. For Pareto’s Law to be true for largeincomes, what conditions must we impose on
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