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Integration

This document is a worksheet from Addis Ababa Science and Technology University focusing on integration in applied mathematics. It contains various problems that require the evaluation of integrals, including both definite and improper integrals, as well as finding volumes of solids of revolution and lengths of functions. The worksheet is structured into sections with multiple integrals to solve.

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Endalew Dejene
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68 views1 page

Integration

This document is a worksheet from Addis Ababa Science and Technology University focusing on integration in applied mathematics. It contains various problems that require the evaluation of integrals, including both definite and improper integrals, as well as finding volumes of solids of revolution and lengths of functions. The worksheet is structured into sections with multiple integrals to solve.

Uploaded by

Endalew Dejene
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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ADDIS ABABA SCIENCE AND TECHNOLOGOY UNIVERSITY

COLLEGE OF NATURAL AND SOCIAL SCIENCES


DEPARTMENT OF MATHEMATICS
APPLIED MATHEMATICS I
WORKSHEET ON INTEGRATION
1. Evaluate the following integrals
𝑥
a) 𝑑𝑥 f) 𝑠𝑖𝑛5 (𝑥)𝑑𝑥
1−4𝑥 2

𝑙𝑛𝑥
b) 𝑑𝑥 g) 𝑐𝑜𝑠 7 (𝑥)𝑑𝑥
𝑥

c) 𝑒 2𝑥 sin(4𝑥) 𝑑𝑥 h) 𝑠𝑖𝑛4 (𝑥)𝑐𝑜𝑠 3 (𝑥)𝑑𝑥

d) 𝑒 2𝑥 cos(6𝑥) 𝑑𝑥 i) sin(7𝑥)cos(9𝑥)𝑑𝑥

3
e) 𝑙𝑛𝑥 𝑑𝑥 j) 𝑠𝑒𝑐 5 (𝑥)𝑑𝑥

2. Evaluate the following integrals


5
𝑥2 𝑥2 1
a) 2
𝑑𝑥 c) 𝑑𝑥 e) 4
0
𝑑𝑥
4−𝑥 𝑥 2 −4 25−4𝑥 2

𝑥2 3 1 2𝑥−3
b) 𝑑𝑥1 d) 1
𝑑𝑥 f) 𝑑𝑥
4+𝑥 2 1+𝑥 2 4𝑥−𝑥 2 −3

3. Find each of the following integrals

𝑥 2 −4 𝑥 2 −1 𝑒𝑥
a) 2 𝑑𝑥 c) 𝑑𝑥 e) 𝑑𝑥
(𝑥) 𝑥−1 𝑥 2 +3𝑥+4 𝑒 2𝑥 +3𝑒 𝑥 +2

6 6 2𝑥 3 +𝑥 2 +12 0 𝑥 2 +𝑥+1
b) 2 𝑥 2 +2𝑥−3
𝑑𝑥 d) 𝑑𝑥 f) −1
𝑑𝑥
𝑥 2 −4 𝑥 2 +1

4. Find the value of the following improper integrals if it converges


𝜋
∞ 𝑒− 𝑥 ∞
a) 1
𝑑𝑥 c) 2
0
𝑠𝑒𝑐 5 (𝜃) 𝑑𝜃 e) −∞
𝑥 𝑒 −𝑥 𝑑𝑥
𝑥

3 1 2 𝑥 ∞ 1
b) 1 3 𝑑𝑥 d) 0
𝑑𝑥 f) 1 ( 𝑥) 𝑥+1
𝑑𝑥
𝑥−1 2 𝑥 2 −2

5. Let R be the region between the graph of the function 𝑦 = 𝑥 2 and the 𝑥 - axis on 0, 1 . Find the

volume V of the solid obtained by revolving R about a) 𝑥 – axis b) 𝑦 – axis

𝜋
6. Let R be the region between the graph of the functions 𝑓 𝑥 = sin(𝑥) and 𝑔 𝑥 = cos(𝑥) on 0, .
4

Find the volume V of the solid generated by revolving R about the a) 𝑥 – axis b) 𝑦 – axis

7. Find the length ℓ of the function 𝑓 𝑥 = 4 − 2 x 3 on 0, 4 .

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