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Page 1 Math 152, Spring 2008-copyright Joe Kahlig 1. (7 points) Find the function y that will solve the differential equation. 2xy ′ + y = 6x where x > 0 2. (8 points) Find the function y that will solve the initial value problem. sec(x) dy = ey+sin(x) where y(0) = 0 dx 3. (9 points) Find the arc length of the curve x = t3 , y = 1.5t2 for 0 ≤ t ≤ 3 4. (6 points) Use the Comparison Theorem to determine whether the integral Z∞ x3 1 dx is convergent or + x2 1 divergent. Be sure to justify your answer. 5. (7 points) Using the error bound for Simpson’s Rule, what is the smallest value of n that guarantees the R2 approximation ln(x) dx is accurate to within 303 × 10−8 ? 1 6. (8 points) The speedometer reading v on a car was observed at 2-minute intervals and recorded in the chart below. Use the Trapezoidal Rule with n = 5 to estimate the distance traveled by the car during these 10 minutes; 1 i.e. hour. 6 t(min) 0 2 4 6 8 10 v(mi/h) 40 45 52 56 57 56 7. (8 points) Set up the integral that will give the area of the surface when the curve x = 2y 4 + y from the point (3, 1) to the point (34, 2) is rotated about the y-axis. You do not need to compute this integral. 8. (9 points) A tank initially contains 100L of fresh water. A solution containing 0.4 g/L of a chemical enters at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at the same rate. (A) Find an expression for y(t), the amount of the chemical (in grams) in the tank after t minutes. (B) How much of the chemical is in the tank after 20 minutes? 9. (6 points) A cube with 0.5-meter-long sides is sitting on the bottom of a tank in which the water level is 2 meters deep. Find the hydrostatic force on the top of the cube. Assume that the bottom of the tank is level. 10. (10 points) An Aquarium in a restaurant has an odd-shaped vertical window at its end that opens to the public view. The water level in the aquarium is level with the top of the window, which ranges from the bottom of the aquarium at its middle to 1 foot high at the top. The window is bounded by the curve y = x6 , −1 ≤ x ≤ 1, and the line y = 1. Find the hydrostatic force pushing against this vertical window. The weight density δ = ρg = 62.5 lb/ft3 . 11. (3 points) The following masses m1 = 5 and m2 = 8 are located at the points P1 (−1, 5) and P2 (6, 2). Compute the moment Mx . 12. (10 points) Find the centroid (center of mass of a region of constant density ρ = 1) of the region in the xy-plane bounded by the curves. √ y=0 x=1 x=5 y= 3x Re 1 dx is convergent or divergent. If it converges, determine x(ln x)2 1 its value. If it diverges, explain in detail why this is so. 13. (9 points) Determine whether the integral Check the back of the page for more problems.