MATH homework (( need to be handwritten ))
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math_10.pdf
La Roche College Math 1033: Analytic Geometry and Calculus II
Summer 2016 Dr. Ryan O’Grady
Midterm Exam (Mock)
INSTRUCTIONS:
• Read each problem carefully.
• Answer each question as best you can.
• Be sure to show all details.
• If you need more space please use the back of the sheet.
• Good luck!
Name:
YOU MUST SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT.
Question: 1 2 3 4 5 Total
Points: 15 15 15 15 15 75
Score:
1. (15 points) Choose TWO of the integrals below and evaluate on the next page(s).
1.
∫ ln(x) dx
2.
∫ 1
1 + 4x2 dx
3.
∫ cos(x)e5x dx
4.
∫ cos2 x dx
5.
∫ 6y1.5 + sec2(y) + 7 dy
6.
∫ cos2(x) sin3(x) dx
7.
∫ te−t dt
8.
∫ x
√ 1 −x2
dx
9.
∫ x sec2 x dx
10.
∫ x
x + 23 dx
Page 2
Space
Page 3
2. (15 points) Consider the region bounded by the curve y = 2x2 −x3 and x-axis on [0, 2]. Determine the volume of the solid obtained when this region is rotated about the y-axis.
Page 4
3. Assume that the formula∫ (ln x)
n dx = x (ln x)
n −n ∫
(ln x) n−1
dx (1)
is true for all positive integers n.
(a) (5 points) Use equation (1) to compute∫ 1 0
(ln x) 2 dx.
HINT: This means set n = 2.
Page 5
(b) (5 points) Use equation (1) to compute∫ 1 0
(ln x) 3 dx.
HINT: This means set n = 3.
(c) (5 points) Make a conjecture on the value of∫ 1 0
(ln x) n dx.
Page 6
4. (15 points) You have just accepted a position with the Department of Defense working on ballistics. Your first assignment is as follows: Consider the region R bounded by the curve y =
√ 4 −x2 for 0 ≤ x ≤ 1, the line y = 0, and the line x = 0. Determine the
surface area of the solid obtained by revolving R about the x axis.
Page 7
5. (a) (10 points) Compute the integral ∫ T 0
xe−x dx
where T > 0.
(b) (5 points) Compute
lim T→∞
∫ T 0
xe−x dx
Page 8
