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The University of British Columbia Sessional Examinations - April 2012
Mathematics 101 Integral Calculus with Applications to Physical Sciences and Engineering
Closed book examination Time: 2.5 hours
Last Name: First Name:
Student Number: Instructor’s Name:
Signature: Section Number:
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBC-card for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishonest practices shall be immedi- ately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid de- vices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material form the examination room without permission of the invigilator.
- Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Total 100
April 2012 Mathematics 101 Page 2 of 12 pages
Marks
[30] 1. Short-Answer Questions. Put your answer in the box provided. Simplify your answer as much as possible. Full marks will be awarded for a correct answer placed in the box. Show your work, for part marks. Each question is worth 3 marks, but not all questions are of equal difficulty.
(a) Evaluate
1
x^2 + 2 x^2
dx.
Answer
(b) Evaluate
(^) π/ 2
0
cos x 1 + sin x
dx. Remember to simplify your answer completely.
Answer
(c) Evaluate
− 2012
x^1 /^3 cos x dx.
Answer
April 2012 Mathematics 101 Page 4 of 12 pages
(g) Let Rn =
^ n
i=
iei/n n^2
. Express limn→∞ Rn as a definite integral. Do not evaluate this
integral. Answer
(h) Express 2. 656565 · · · as a rational number, i.e. in the form p/q where p and q are integers. Answer
(i) Evaluate lim x→ 0
sin x − x + x^3 / 6 sin(x^5 )
Answer
(j) Using a Maclaurin series, the number a = 1/ 5 − 1 /7+1/18 is found to be an approximation for I =
0 x
(^4) e−x^2 dx. Give the best upper bound you can for |I − a|.
Answer
April 2012 Mathematics 101 Page 5 of 12 pages
Full-Solution Problems. In questions 2–8, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions.
[10] 2. Let R be the bounded region that lies between the curve y = 4 − (x − 1)^2 and the y = x + 1.
(a) [6] Sketch R and find its area.
(b) [4] Write down a definite integral giving the volume of the region obtained by rotating R about the line y = 5. Do not evaluate this integral. Answer
April 2012 Mathematics 101 Page 7 of 12 pages
(c) [5] Evaluate
4 x + 8 (x − 2)(x^2 + 4)
dx
Answer
(d) [5] Determine, with explanation, whether the improper integral
−∞
x x^2 + 1
dx converges or diverges.
April 2012 Mathematics 101 Page 8 of 12 pages
[12] 4.
(a) [4] Determine, with explanation, whether the series
^ ∞
n=
n^2 − sin n n^6 + n^2
converges or diverges.
(b) [4] Determine, with explanation, whether the series
^ ∞
n=
(−1)n(2n)! (n^2 + 1)(n!)^2
converges abso-
lutely, converges conditionally, or diverges.
(c) [4] Determine, with explanation, whether the series
^ ∞
n=
(−1)n n(ln n)^101
converges absolutely,
converges conditionally, or diverges.
April 2012 Mathematics 101 Page 10 of 12 pages
[8] 6. A tank in the shape of a hemispherical bowl of radius 3 m, with an outlet that rises 2 m above its top (see the diagram below), is full of water. Using the fact that the density of water is 1000 kg/m^3 , find the work (in Joules) required to pump all the water out of the outlet. You may use the value g = 9.8 m/s^2 for the acceleration due to gravity. You do not need to simplify your answer, but you must completely evaluate any integral(s) that arise. Answer
3 m
2 m
April 2012 Mathematics 101 Page 11 of 12 pages
[6] 7. Let I =
1 (1/x)^ dx. (a) [2] Write down the trapezoidal approximation T 4 for I. You do not need to simplify your answer.
(b) [2] Write down the Simpson’s approximation S 4 for I. You do not need to simplify your answer.
(c) [2] Without computing I, find an upper bound for |I − S 4 |. You may use the fact that if |f (4)(x)| ≤ K on the interval [a, b], then the error in using Sn to approximate
(^) b a f^ (x)^ dx has absolute value less than or equal to K(b − a)^5 / 180 n^4.
