Calculus II PLTLFall 2014Worksheet 1These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Pairs) Each of the following definite integrals represents an area that can be computedusing facts from geometry. For each integral, sketch the region des cribed and give the areaof the region.(a)R514 dx (b)R513x dx (c)R51|4 − 3x|dx (d)R20√4 − x2dx2) (Round Robin) Find the area of the region enclos ed by the curves pictured below.3) (Scribe) Find the a rea of the region in the firs t quadrant bounded on the left by they-axis and on the right by the curves y = sin x and y = cos x.4) (Round Robin) Let f(x) = −x2.(a) Graph f on the interval [0, 1].(b) Partition the interval into four subintervals of equal length.(c) Add to your sketch the rectangles associ a ted with the Riemann sumP4k=1f(ck)∆xk, where ckis the left-hand endpoint of the kthinterval.(d) How is the integralR10−x2dx related to the limit limn→∞Pnk=1f(ck)∆xk?(e) Using what you have lear ned abou t forming a Riemann sum using left endpoints,find a function g and numbers a and b such thatZbag(x) dx = limn→∞nXk=1g(ck)∆xk= limn→∞(4)2/34n+4 +4n2/34n+ ··· +4 + (n − 1)4n2/34n5) (Scribe) Eval uateR80t√t + 1 dt.6) (Pairs) Consider the fun ction g(x) =Rcos x0x2cos(2t) dt.(a) Eval uate the integral to get an a lternate expression for g(x), and then find g0(x).(b) Use the Fu ndam ental Theorem of Calculus (and the Chain Rule) to find g0(x)from the original expression, then compare with your previous
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