Practice Set #2—Math 101, Section 213
Surname (Last Name)
Given Name
Student Number
Problem Out of Score Problem Out of Score
1
6
5
8
2
6
6
8
3
8
7
8
4
6
Total
50
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
1a. [3 pts] Compute the following limit
n
X
i4
lim
n→∞
n5
i=1
1b. [3 pts] Write the general form partial fraction decomposition corresponding to the rational
function
1
(x + 72) · (x + 1)2 · (x2 + 9)
DO NOT find the coefficients.
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
2a. [3 pts] Express the following limit
n
X
iπ
π
lim
· tan( )
n→∞
4n
4n
i=1
as a definite integral and then evaluate it.
2b. [3 pts] Find the derivative of the function
Z x3 √
f (x) =
t sin t
x
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
3a. [5 pts] Find the centroid of the region bounded by the curve y = 1 + sin x, the x-axis
(y=0) and the lines x = ± π2 .
3b. [3 pts] Using n = 4 and Simpson’s Rule, approximate the value of
Z 4
1
dx
3
0 x +1
You can leave your answer as a sum of fractions, don’t simplify.
Problems 1–4 are short-answer questions: put a box around your final answer, but no credit will be
given for the answer without the correct accompanying work.
4. [6 pts] Integrals
(a) Evaluate
π/4
Z
cos4 x dx
0
(b) Evaluate
2015
Z
x1/3 cos x dx
−2015
(c) Evaluate
Z
∞
ln x
dx
x101
1
if it converges. If it doesn’t, explain why.
(d) Evaluate
∞
Z
x + sin x
dx
1 + x2
2
if it converges. If it doesn’t, explain why.
(e) Evaluate
Z
x arctan x dx
(f) For what values of p does
Z
e
converge?
∞
1
dx
x(ln x)p
Problems 5–7 are long-answer: give complete arguments and explanations for all your calculations—
answers without justifications will not be marked.
5a. [4 pts] Express as a definite integral the volume of the solid obtained by rotating the region
above the x-axis, below y = sinx x and between x = π2 and x = π about the y-axis (but don’t
evaluate it).
5b. [4 pts] Evaluate the volume obtained by rotating the region under the curve y = sin x,
above the x-axis, between x = 0 and x = π about the line y = 1.
6.
6a. [4 pts] A cable that weighs 2lb/ft is used to lift 800lb of coal up a mine shaft 500ft deep.
Find the total work done. It is recommended you write everything as a limit of Riemann
Sums first.
6b. [4 pts] Evaluate the integrals
(a)
Z
√
x
dx
1 − x4
(b)
Z
1
√
dx
x2 + 2x + 5
(c)
Z
cos3 x sin4 x dx
(d)
Z
p
cos( (x) dx
7a. [2 pts] Solve the IVT
y 0 = xy 2 , y(0) = 1
7b. [6 pts] An open metal tank has two ends which area isosceles triangles with vertex at the
bottom, two sides which are rectangular, and an open top. The tank is 1m wide, 2m deep,
10m long and full of water (the density of which is 1000kg/m3 ). Express the work required
to pump the water out of the tank if it is pumped out of the top edge as an integral (but do
not evaluate it). It is recommended you write everything as a limit of Riemann Sums first.