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Berkeley MATH 1B - Math 1B PDP Worksheet

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Rob Bayer Math 1B PDP Worksheet February 5, 2009Instructions1. Introduce yourselves!2. Find some blackboard space, a piece of chalk, and decide who will be your first scribe.3. Do the problems below, having a different person be the scribe for each one.Improper Integrals1. Which of the following integrals are improper? For those that are improper, show how you would break themup into the limit of proper integrals and then decide whether they converge or diverge. Do not use thecomparison test.(a)Zπ/20sec xdx(b)Z3−31x2+ 1dx(c)Z∞4e−y/2dy(d)Z∞−∞1x3dx(e)Z10dy4y − 1(f)Z1−1exex− 1dx(g)Z∞0sin tdt2. Consider the integralZ1−11x4/3.(a) Explain why this integral is improper and show it diverges.(b) What would you get if you “forgot” that it was improper and just evaluated its anti-derivative at theendpoints? Why does this answer not make any sense?3.Why we do limt→∞Zt0+ lims→−∞Z0s(a) Consider the integralR∞−∞2xdx. Show that this integral diverges.(b) Evaluate each of the following and explain why from a naive standpoint they seem likeZ∞−∞i. limt→∞Zt−t2xdxii. limt→∞Zt+1−t2xdxiii. limt→∞Z√t2+1−t2xdx(c) Convince yourself that part (b) really does show thatZ∞−∞= limt→∞Zt−tis not a good definition.4. FindZ∞0ln x1 + x2dxIntegration Practice1. Find each of the following integrals. Be sure to work as a group so everyone knows how to do all theseproblems.(a)Zex+exdx(b)Zsec2(sin θ)sec θdθ(c)Z1√x + 1 +√xdx(d)Zln(x + 1)x2dx(e)Zt3+ 1t3− t2dt(f)Zcos4t − sin4tdt(g)Zcos32x sin 2xdx(h)Zdt√et(i)Z1x√x2+ 4dx(j)Z1x√x + 4dx(k)Zx√x2+ 4dx(l)Z14√x +3√x(m)Zln(sec θ) sec2θdθExtra Problems If you finish early, take a stab at these.1. (The Weirstrass Substitution) It turns out that any rational function of sin and cos (and hence, ofsec, tan, csc, cot, etc) can be turned into an ordinary rational function via the substitution t = tan(x2). Let’sexplore why:(a) Show that cosx2=1√t2+ 1and that sinx2=t√t2+ 1. Hint: right triangles(b) Use trig identities to show that cos x =1 − t21 + t2, and that sin x =2t1 + t2(c) Show that dx =21+t2dt(d) Use parts (b) and (c) to findZdx3 sin x − 4 cos xandZsec3xdx2. Find each of the following. While you could use the technique of the previous problem for (a) and (b), youshould try to find a more straightforward way.(a)Zcos x − 1cos x + 1dx (b)Zsin x + cos xsin 2xdx(c)Zdx√x +3√x3. Let S be a sphere of radius 1. Now take a plane a distance d < 1 away from the center and slice the sphere intotwo pieces. What is the volume of each piece? Hint: the formula for the volume of a revolved solid isRbaπf


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Berkeley MATH 1B - Math 1B PDP Worksheet

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