Calculus 2

4) Evaluate the improper integral from zero to infinity of f(x)=e*. (10 points) 5) Evaluate the improper integral from negative infinity to infinity of f(x)=1/(1+x]). (10 points) 6) Why is the integral from x=1 to x=3 of f(x)=1/(x-2)" improper? (10 points) 7) Find the area of the region bounded by the graphs of f(x)=5-x and g(x)=x'-3. (10 points) 8) When we find the area between an upper curve f(x) and a lower curve g(x), we integrate f(x)-g(x). How do we know that this result is positive? (5 points) 9) Use the method of disks to rotate a circle of radius r around the x axis and prove the volume formula for a sphere of radius r. (10 points) 10) Describe what circumstances would cause you to use the method of washers rather than the method of disks. (5 points) 11) Use the method of shells to find the volume of the solid obtained by rotating about the y axis the region bounded by y=2x'-x' and y=0. (10 points) 12) Find the length of the arc of y=x" between the points (1,1) and (4,8). (10 points) 13) Find the limit of the sequence a, = { sin(n)}, if it exists. (10 points) 14) Find the sum of the series whose terms are a = 1/5". What kind of series is this? (10 points) 15) Find the sum of the series whose terms are a = {(1)- (1/(n+1));. What kind of series is this? (10 points) 16) Does the series which is the sum of the terms a = 1" converge or diverge? Explain. (10 points)