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MATH 152, Fall 2013 Week In Review Week 2 1. Evaluate e Z √ 1 lnx dx. x 2. Compute the area enclosed by the parabola y = x2 and the straight line y = 2x. 3. If f (x) = Z 0 x2 tan(t2 + 1) dt, then nd f 0 (x). 4. Evaluate the integral 5. Find Z 1 2 R2 √ −2 4 − x2 dx by interpreting it in terms of areas. x dx . x+1 6. Suppose it is given f (5) = 3, f 0 (5) = 2, f (1) = 4, and f 0 (1) = −1. Compute R5 1 xf 00 (x)dx. 7. If g(x) = 10 Z p t2 + 1dt, then nd g 0 (x). ln(x) 8. Find the integral 9. If h(x) = Z 5x cos(x) R x5 ex dx. 2 cos(t2 ) dt, then nd h0 (x). 10. Find the area bounded by the curves x + y 2 = 2 and x + y = 0 . √ 11. Let R denote the region enclosed by the x-axis, the line x = 1, and the curve y = x. Use the method of cylindrical shells to calculate the volume of the solid obtained by rotating the region R about the line x = 1. 12. Find Z √ x3 dx . x2 + 1 13. An animal population is increasin at a rate of 200 + 50t per year ( where t is measured in years). By how much does the animal population increase between the fourth and tenth years ? 14. Find the interval on which the curve Z y= 0 x 1 dt 1 + t + t2 is concave upward . 15. The velocity function ( in meters per second ) is given for a particle moving on a line as follows v(t) = t2 − 2t − 8, 1≤t≤6 Find (a) the displacemenet and (b) the distrance traveled by the particle during the given time interval.