Calculus II PLTL Fall 2014 Worksheet 1 These problems are to be done without the use of a calculator unless otherwise specified 1 Pairs Each of the following definite integrals represents an area that can be computed using facts from geometry For each integral sketch the region described and give the area of the region a R5 1 4 dx b R5 1 3x dx c R5 1 4 3x dx d R2 4 x2 dx 0 2 Round Robin Find the area of the region enclosed by the curves pictured below 3 Scribe Find the area of the region in the first quadrant bounded on the left by the y 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 associated with the Riemann sum P4 th interval k 1 f ck xk where ck is the left hand endpoint of the k R1 Pn d How is the integral 0 x2 dx related to the limit limn k 1 f ck xk e Using what you have learned about forming a Riemann sum using left endpoints find a function g and numbers a and b such that Z b n X g x dx lim g ck xk a n k 1 lim n 5 Scribe Evaluate 2 3 4 4 4 2 3 4 4 2 3 4 4 4 n 1 n n n n n R8 t t 1 dt 0 R cos x 2 6 Pairs Consider the function g x 0 x cos 2t dt a Evaluate the integral to get an alternate expression for g x and then find g 0 x b Use the Fundamental Theorem of Calculus and the Chain Rule to find g 0 x from the original expression then compare with your previous answer
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