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CLS601S - CALCULUS 2 - 1ST OPP - JUNE 2023 |
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nAmlBIA UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACULTYOF HEALTH,NATURALRESOURCESAND APPLIEDSCIENCES
SCHOOLOF NATURALAND APPLIEDSCIENCES
DEPARTMENT OF MATHEMATICS, STATISTICS AND ACTUARIAL SCIENCE
QUALIFICATION: Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
COURSE CODE: CLS601S
LEVEL: 6
COURSE NAME: CALCULUS 2
SESSION: JUNE 2023
DURATION: 180 MINUTES
PAPER: THEORY
MARKS: 100
EXAMINERS
MODERATOR:
FIRSTOPPORTUNITYQUESTION PAPER
MR BENSON OBABUEKI
DR SERGE NEOSSI-NGUETCHUE
DR DAVID IIYAMBO
INSTRUCTIONS
1. Answer ALL questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil.
PERMISSIBLEMATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 2 PAGES (excluding this front page)
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Question 1 {35 marks)
Determine the following indefinite integrals using the indicated techniques:
1.1
J2
3xex dx by substitution.
(5)
1.2
Je3-' sin 2xdx by parts. Start with u = sin 2x.
{9}
f 1.3
by trigonometric substitution.
(5)
36x +1
f 1..4
sin 4 ycos 3 ydy by any (simple} method.
(7)
1.5
f-.-dx using the t-formula.
smx
(9)
Question 2 (14 marks)
2.1 Determine the area enclosed by y = x 2 -9 and y = 9- x 2 •
(5)
2.2 What is the arc length of y = In sec x in the interval O x {-? (Hint: sec x is positive in
the given interval}
(5)
2.3 Calculate the volume of the solid generated if y = 2x is rotated about the x-axis
through a complete revolution, 0 x 4.
(4)
Question 3 {16 marks)
fI
The definite integral ex2+3dx is to be estimated using the Simpson Rule, correct to within an
0
error of 0.7%.
3.1 Determine the number of subintervals needed.
(8)
3.2 Use n =8 to estimate the given definite integral to within an error of 0.7%.
(8)
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Question 4 (13 marks)
4.1
=In: f Given that f(x)
=I
4
1 11
x1
,
determine
the
definite
integral
f(x)dx.
O
(6)
4.2 Use Taylor series to approximate fo°1s"in(x 4 )dx to within an error of 10-20 • (Hints: (1)
L - cc ( 1)11 211+!
sin u =
u ; (2) The alternating series error estimation rule allows you to
11=0 (2n + 1)!
truncate from the first term that has an absolute value less than the error limit.) (7)
Question 5 (11 marks)
The curve y = x-x
2
,
between
x = 0 and x = 2, rotates about the x-axis through a complete
revolution. Determine the centre of gravity of the solid so formed.
(11)
Question 6 (11 marks)
A curve is defined by the parametric equations x=0-sin0 and y=l-cos0. Determine the
area generated by the curve between 0 = 0 and 0 = 2n, when rotated completely about the x
-axis.
(11)
End of Paper
Total marks: 100%
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