MA 126, Calculus 2 UAB, Spring 2002
TEST 2
Duration 105min; Max. Points: 40
Make sure to show all your work and underline the final results of each problem. Write your
name on this sheet and use it as a cover page when you turn in your work. Do not write
your results on this paper. Good luck!
1. (6 pts.) (a) Use the Simpson rule with two subintervals to find an approximation for
Z1
0
e−x2dx.
(b) Estimate the error in part (a). You may use that the 4th derivative of f(x) = e−x2is
given by f(4)(x) = (12 −48x2+ 16x4)e−x2.
(c) How large do you need to choose the number nof subintervals to guarantee an accuracy
of 0.001 for the estimate by the Simpson rule? (If you did not solve (b), work with K= 90.)
2. (4 pts.) (a) Set up the integral for the length of the curve given by the parametric
equations
x=√2 cos(t), y = sin(t),0≤t≤π.
(b) Use the trapezoidal rule with n= 2 subintervals to find an approximation for the integral
you obtained in part (a).
3. (4 pts.) Sketch the region bounded by the graph of y= 1/x and the lines
y= 0, x = 1, x = 2.
Compute the volume of the solid obtained by rotating this region about the x-axis.
4. (6 pts.) Which of the following improper integrals are convergent? Evaluate the ones
which are.
(a)Z∞
0
cos(x)dx
(b)Z∞
e
ln(x)
xdx
1
