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Practice exam 3. Name: Calculators allowed but I strongly urge you not to use one! Remember that points are given for the steps that you perform, not just the answer. 1. Calculate each of the following indenite integrals: (a) (b) (c) Z Z Z x5 , 3x7 sin dx d (t3 + 4) (t + 1) dt Z sin x (2 + cos x)2 dx (b) Check your answer to part (a) by dierentiating. 2. (a) Calculate (c) Calculate Z cos(2x) sin(x) dx (d) Check your answer to part (c) by dierentiating. 3. Solve the dierential equation dy dx = x sec y. R 4. (a) Find the Riemann sum for the integral 06 x + 4 dx using left endpoints and n = 3. (b) Do the same for the general case where the interval is divided into n pieces of length x = 6=n. R (c) Compute the integral 06 x + 4 dx by computing the limit as n ! 1 of your answer to (b); you will need the formula =1 i = 21 n(n + 1). (d) State the Fundamental Theorem of Calculus: R (e) Compute the integral 06 x + 4 dx using the fundamental theorem of calculus. 5. Find the area under the curve y = cos x between x = 0 and x = =2. n i 1