Review problems for the fourth midterm
The fourth midterm will be held on Tuesday, November 13, in class. It will cover the lectures, homework
assignments, and text associated with Chapter 4 in our book. Following are some problems which are
representative of what may appear on the test. That does not mean that the test questions will look exactly
like these, but they should give you a good indication of what’s most important to know.
Here are the questions, in roughly “chronological” order:
1. Find the antiderivatives of the following functions:
a) Z(x3+ 2) dx b) Z4
x2dx c) Z2
xdx
d) Z3 sec2x dx e) Zx2+ 4
xdx f) Zex(1 −e−x)dx
g) Z2
1 + x2dx h) Z3
√1−x2dx i) Z(ln 2)2xdx
2. Suppose an object moves along the x-axis with acceleration function a(t) = 6 −t, with initial velocity
v(0) = 1, and initial position s(0) = −1. Determine the position function s(t) for the object.
3. Find the sums:
(a)
100
X
i=1
(i2−1) (b)
100
X
i=1
(i2+ 2i)
4. For the function f(x) = x2−2xon the interval [0,2], list the evaluation points for the Midpoint Rule
with n= 4, sketch the function and the approximating rectangles, and evaluate the Riemann sum.
5. Approximate the area under the curve y=1
xover the interval [1,2] using 5 rectangles and the right-
endpoint evaluation rule.
6. Write a Riemann sum that approximates
Z1
0
2x2dx
using nrectangles. Do not use any shortcut formulas to simplify your sum.
7. Suppose an integral has a Riemann sum approximation Anusing nrectangles, which, when simplified
using shortcut formulas, becomes
An=(n+ 1)(n−1)
3n2.
What is the value of the integral?
8. Find the definite integrals:
a) Z1
0
(x4−7) dx b) Z3
1
1
x2dx c) Z4
1
2
xdx
d) Zπ/4
0
3 sec xtan x dx e) Z3
2
x2−1
x+ 1 dx f ) Z1
0
e2xdx
g) Z1
−1
2
1 + x2dx h) Zπ/2
0
cos(2x)dx i) Z1
0
(ln 2)2xdx
9. Find the average value of the function f(x) = exon the interval [0,2].
10. Find the average value of the function f(x) = sin xon the interval [0, π].
