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Integral Calculus Exam

The document is an examination paper for the Bachelor of Education degree at Uganda Christian University, focusing on Integral Calculus. It contains five questions covering topics such as evaluating integrals, mean value theorems, the fundamental theorem of calculus, improper integrals, and finding areas and arc lengths. Students are instructed to attempt any four questions within a duration of three hours.

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59 views3 pages

Integral Calculus Exam

The document is an examination paper for the Bachelor of Education degree at Uganda Christian University, focusing on Integral Calculus. It contains five questions covering topics such as evaluating integrals, mean value theorems, the fundamental theorem of calculus, improper integrals, and finding areas and arc lengths. Students are instructed to attempt any four questions within a duration of three hours.

Uploaded by

winnliky
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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UGANDA CHRISTIAN UNIVERSITY

BACHELOR OF EDUCATION DEGREE (SECONDARY)


MATH(S1202) INTEGRAL CALCULUS-2021
Duration: 3hours
Instructions: Attempt any four questions in this paper
Question one:
(a) Evaluate the integrals as a limit of Riemann sums.

1
(i) ∫0 𝑥 2 𝑑𝑥 (09marks)

3
(ii) ∫1 (3𝑥 2 + 1)𝑑𝑥 (08marks)

𝑥 3 −𝑥 2 −4𝑥+1
(b) Use partial fractions to evaluate∫ 𝑑𝑥 (08marks)
𝑥 2 −4

Question two:

(a) Find the mean value of the function 𝑓(𝑥) = 3𝑥 2 − 2𝑥 on the interval [1,4].
(06marks)
(b) Find the mean value of the function 𝑓(𝑥) = |𝑥 − 1| on the interval [-1,2].
(07marks)
2
(c) (i) Let 𝑓(𝑥) = 3𝑥 + 2𝑥. If 𝐹 is an anti-derivative of 𝑓 on the interval 𝐼 and 𝐹(1) = 3.
Find 𝐹(𝑥). (06marks)
(i) Find the upper bound and the lower bound of the integral
3
∫1 √𝑥 3 + 1 𝑑𝑥 (06marks)

Question three
(a) State the fundamental theorem of calculus. (04marks

(b) Use the fundamental theorem of calculus to evaluate 𝐹 ′ (𝑥) given that

𝑥2
(i) 𝐹(𝑥) = ∫0 𝑥√𝑥 2 − 1 𝑑𝑥 (08marks)

𝑠𝑖𝑛𝑥
(ii) 𝐹(𝑥) = ∫0 √1 − 𝑡 2 𝑑𝑡 (08marks)
(c) State and prove the mean value theorem for integral calculus. (05marks)

Question four

(a) Evaluate the following integrals

𝑥2
(i) ∫ 1+𝑥 2 𝑑𝑥 (05marks)

(ii) ∫ 𝑒 𝑥 𝑐𝑜𝑠𝑥𝑑𝑥 (05marks)

𝑑𝑥
(iii) ∫ √16−9𝑥 2 (05marks)

(b)

3 5+𝑥
(i) Find ∫2 𝑑𝑥 correct to three significant figures. (05marks)
(1−𝑥)(5+𝑥 2 )

(ii) Prove the reduction formula


1 𝑛−1
∫ 𝑐𝑜𝑠 𝑛 𝑥𝑑𝑥 = 𝑛 𝑐𝑜𝑠 𝑛−1 𝑥𝑠𝑖𝑛𝑥 + 𝑛 ∫ 𝑐𝑜𝑠 𝑛−2 𝑥𝑑𝑥
Hint: use integration by parts, let 𝑢 = 𝑐𝑜𝑠 𝑛−1 𝑥 𝑑𝑣 = 𝑐𝑜𝑠𝑥 (05marks)
Question five
(a) state the type of integral whether its 𝑇𝑦𝑝𝑒 𝐼 or 𝑇𝑦𝑝𝑒 𝐼𝐼

∞
(i) ∫0 𝑒 −𝑥 𝑑𝑥
03marks
∞ 𝑑𝑥
(ii) ∫0 𝑥−1

(b) Evaluate the following improper integrals and state whether they are convergent or
divergent

∞ 𝑑𝑥
(i) ∫1 04marks
𝑥

+∞ 𝑑𝑥
(ii) ∫−∞ ( 06marks)
1+𝑥 2

(c) (i) Sketch the region bounded by the given curves and find the area between them
𝑦 = 𝑥2 and 𝑥 = 𝑦 2 ( 08marks)
2
(𝑖𝑖) find the arc length 𝑦 = 𝑥 3 + 5 from 𝑥 = 1 to 𝑥 = 8 (04marks)

END

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