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MAT 192, Final Exam, Fall 2016Name ______________________________________________Instructions: Show all work. Answers without work required to obtain the solution will not receive full credit. Some questions may contain multiple parts: be sure to answer all of them. Give exact answers unless specifically asked to estimate.Find the volume of the solid of revolution defined by y=14x2,y=5-x2 revolved around the x-axis. (6 points)Set up an integral to find the volume of the solid of revolution defined by x2+4y2=4, revolved around the line x=2. You do not need to integrate it. (5 points)Find the volume of the solid of revolution bounded by y=x2,y=6x=2x2 around the y-axis. (6 points)Set up an integral to find the volume of the solid of revolution defined by x=y-12,x-y=1, revolved around the line x=-1. You do not need to integrate. (5 points)Set up an integral to find the surface area of the surface of the surface of revolution defined by y=xex on the interval [0,1] revolved around the x-axis. (5 points)A spherical tank with a radius of 3 meters is full of water. Find the work done pumping all the water out of the top of the tank. Recall that the density of water is 1000 kg/m3. (7 points)A 10-ft chain weighs 25 lbs and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling. (5 points)Find the average value of the function ft=esintcos?(t) on the interval 0,π2. Use your calculator to find c so that fc=f. (5 points)Set up the three integrals needed to find the centroid of the region bounded by y=x3, x+y=2, y=0 with constant density. Sketch the region. You do not need to integrate. (6 points)Find the solution to y'=xysinxy+1, y0=1. (5 points)Match the graphs to the parametric equations that generated them. (3 points each)c. d. e. x=lntcost, y=sintiv. x=6sin3t, y=4costx=sect,y=tan?(8t)v. x=sin3t,y=3costx=cosht, y=t2-3sinh(t)vi. x=sin2t+cost,y=tcostFind the equation of the tangent line to the graph x=t-t-1,y=1+t2 at t=1. (5 points)Set up the integral to find the length of arc of the curve x=t4-2t3-2t2, y=t3-1 on the interval [0,2]. You do not need to integrate. (5 points)Set up the integral to find the area inside r=3+2cosθ and r=3+2sinθ. (5 points)Sketch the graph of the curve r=1+2cosθ. (6 points)Write the sequence 12,-43,94,-165,256,… as a formula for an in terms of n. (4 points)Match the equation to the graph of the polar curve. (3 points each)d. g. e. f. h. r=1-cosθv. r=1-45sin2θr=lnθvi. r=esinθ-2cos?(4θ)r=sin6sinθvii. r=eθ/10sinθr=sin2(4θ)+cos?(4θ)viii. r=2cosθ-secθDetermine whether the sequence converges or diverges. If it converges, what does it converge to? (4 points each)an=n3n+1b. an=e2n/(n+2)Find the sum of the series, if it exists. (4 points each)n=1∞3n+1πnc. n=1∞(e1/n-e1/(n+1)) n=3∞3n4For each series select an appropriate test to determine convergence or divergence of the series. (5 points each)n=1∞n22n-1-5nd. n=2∞1nlnnn=1∞sin2n1+2ne. k=2∞klnkk+13n=1∞-1ncoshnDetermine the interval of convergence of the radius of convergence for the power series n=1∞n2x-1n5n. Be sure to check the endpoints of the interval. (5 points)Find a power series for the function fx=23-x. (5 points)Use a power series to evaluate t1+t5dt. (5 points)Rewrite x5+3x4-2x as a Taylor series centered at c=1. (5 points)nn!0123456Find a Maclaurin series for fx=x2ln1+x3. Use the table of Maclaurin series included at the end of the exam. Graph the first 4 (non-zero) terms on the same graph as the original function. (5 points)Find the area of the region bounded by x=y2-4y and x=2y-y2. Sketch the region. (6 points)Which method of integration should be used to evaluate each integral? You do not need to integrate. If you are using a substitution method, state the substitution used. (3 points each)cosxln(sinx)dxd. cos2θsin2θdθx5e-x3dxe. x2+2xdxtt4+2dtf. x3-4x-10x2-x-6dxIntegrate. (5 points each)sin8xcos5xdxdxcosx-1Apply partial fractions to the expression 1x2-4x2-x-6x2+1x3. You do not need to integrate or solve for the coefficients. (5 points)Use Simpson’s Rule to estimate 0211+x5dx with n=4. (6 points)Determine if the improper integral 1∞lnxxdx converge or diverge. Sketch the graph of the region to find all points of discontinuity insider the interval. If it converges, evaluate it. (5 points)Some useful formulas:d2ydx2=ddtdydxdxdtdydx=rθcosθ+r'θsinθ-rθsinθ+r'θcosθs=αβr2+drdθ2dθn=1∞1n2=π26, n=1∞1/n4=π490fx=n=0∞fncx-cnn!Rnx≤maxfn+1zn+1!xn+1arcsinx=x+x32?3+1?3x52?4?5?7+…+2n!x2n+12nn!22n+1 -1<x<1 a3-b3=a-ba2+ab+b2cos2t=121+cos2tsin2t=2sintcost,cos2t=cos2t-sin2tcosacosb=12cosa+b+cosa-bsinasinb=12cosa-b-cosa+bsinacosb=12sina+b+sina-bTrapezoidal Rule Error:|E|≤b-a312n2maxf''xSimpson’s Rule Error:|E|≤b-a5180n4maxfIVxSimpson’s Rule: f(x)≈b-a3nfx0+4fx1+2fx2+4fx3+…+4fxn-1+fxnTrapezoidal Rule: f(x)≈b-a2nfx0+2fx1+2fx2+…+2fxn-1+fxnM=ρabfx-gxdxMx=ρab12f2x-g2xdxMy=ρabxfx-gxdxx=MyM, y=MxM ................
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