Math 126 Carter Final Fall 2003
Rules and Instructions. Indicate all work and answers in your blue book. Do not
write on this test sheet. Write your name on only the outside of your blue book. Good luck.
1. Indicate the details of your computations for the following definite, indefinite, and
improper integrals (10 points each).
(a) Rcos2(x)dx
(b) R3
1x5ln (x)dx
(c) R1
x2−2xdx
(d) R∞
1
dx
x1.3
2. State the following (10 points):
(a) the definition of the definite integral of a real valued function that is defined on
a closed inteval [a, b];
(b) the fundamental theorem of calculus.
3. (10 points) Compute the area of the planar region that is bounded between y=√x,
and y=x3for x∈[0,1].
4. (10 points) Use cylindrical shells to compute the volume of the solid obtained by
revolving the portion of the graph of e−x2that is between x= 0 and x= 1 around the
y-axis.
5. (10 points) Consider the cone illustrated. What is the volume of the cone?
•First compute the volumes of cone by using the formula V=π
3r2h.
•Check your result by expressing the volume as an integral.
30m
20m
6. (10 points) Find the first 4 terms in the MacLaurin series expansion for the function
f(x) = tan (x).
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