University of California, Riverside
Department of Mathematics
Midterm Exam
Mathematics 9B - First Year of Calculus
Sample 1 (Rev. 090119)
Instructions: This exam has a total of 120 points. You have 50 minutes.
You must show all your work to receive full credit. You may use any result
done in class. The points attached to each problem are indicated beside the
problem. You are not allowed books, notes, or calculators. Answers should
be written as
√
2 as opposed to 1.4142135....
1. Let f (x) = 1− x2.
(a) (8 points) Compute the left-hand Riemann sum approximation of∫ 3
0
f (x) dx with n = 3 boxes.
(b) (8 points) Compute the right-hand Riemann sum approximation
of
∫ 3
0
f (x) dx with n = 3 boxes.
(c) (8 points) Express
∫ 3
0
f (x) dx as a limit of right-hand Riemann
sums (as in the definition of the definite integral). Do not evaluate
the limit.
2. Evaluate the indefinite and definite integrals.
(a) (12 points) ∫
x2
√
1 + x3 dx
(b) (12 points) ∫ pi
2
pi
4
cos(x)
sin2(x)
dx
1
3. Microtutorial (24 points) Assume the brakes of a certain automobile
produce a constant deceleration of 22ft/s2. If the car is traveling at 60
mph (88ft/s) when the brakes are applied, how far will it travel before
coming to a complete stop?
4. Evaluate the indefinite and definite integrals.
(a) (12 points) ∫
x2ex dx
(b) (12 points) ∫ e
1
x3 lnx dx
5. (24 points) Evaluate the integral:∫
sin3 x cos2 x dx