Name: ________________________ Class: ___________________ Date: __________ Final Exam for Math 193 1. a) Evaluate the integral. 20xe 5x dx b) Evaluate the integral. 81 t ln tdx 1 c) Evaluate the integral. 2 sin3 cos 2 d 0 _ d) Evaluate the integral or show that it is divergent. ln x x4 dx 1 e) Evaluate the integral. 4dy y 2 2y 3 2 f) Set up, but do not evaluate, an integral for the length of the curve. y x3 4 x , 0 x 5 1 ID: A Name: ________________________ ID: A 2. a) Solve the differential equation. y 6x 5 y ln y b) Find the solution of the differential equation that satisfies the initial condition y(0) 6 . dy 8x 7 y dx c) A tank contains 200 L of brine with 12 kg of dissolved salt. Pure water enters the tank at a rate of 6 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 20 minutes? 3. a) Find a Cartesian equation for the curve described by the given polar equation. r 3sin b) Find the polar equation for the curve represented by the given Cartesian equation. xy 4 c) Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x t cos t, y t sin t, t d) Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x cos , y sec , 0 2 e) The graph of the following curve is given. Find the area that it encloses. r 2 cos 6 2 Name: ________________________ ID: A 4. a) Find the volume of the solid obtained by rotating the region bounded by y x 3 and x y 3 about a) the x-axis and b) about the line x 1. b) A tank is full of water. Find the work required to pump the water out of the outlet. Round the answer to the nearest thousand. h 1m, r 1 m, d 5 m 5. a) Determine whether the series is convergent or divergent by expressing s n as a telescoping sum. If it is convergent, find its sum. n1 2 2 n 2n b) Test the series for convergence or divergence. 3 k 1 k1 4 2k c) Evaluate the indefinite integral as an infinite series. ex 1 dx x d) Find the Maclaurin series for f(x) using the definition of the Maclaurin series. f x x cos 4x e) Find the radius/interval of convergence. 3