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Definite Integration Tutorial EN1911 (Mathematics 1A)Tutorial – Definite Integration
Evaluate the following definite integrals, give exact answers where possible:
2 3 2 𝑑𝑥
1. ∫0 𝑥 2 𝑑𝑥 2. ∫0 (4 − 𝑥)2 𝑑𝑥 3. ∫1 𝑥2
4 2 9 𝑑𝑝
4. ∫0 3√𝑥 𝑑𝑥 5. ∫1 𝑡 2 + 3𝑡 𝑑𝑡 6. ∫1
√𝑝
𝜋/2 1/2 0.8
7. ∫0 4 cos(𝑥) 𝑑𝑥 8. ∫1/3 cos(2𝑥) 𝑑𝑥 9. ∫0 𝑒 2𝑥 𝑑𝑥
3 𝑑𝑥 𝜋/4 4 𝑤+1
10. ∫2 𝑥
11. ∫𝜋/6 sec 2 (𝜃) 𝑑𝜃 12. ∫1 2√𝑤
𝑑𝑤
1 𝑒 𝑡+𝑒 −𝑡 𝜋/𝜔 4
13. ∫1/2 2
𝑑𝑡 14. ∫0 sin(𝜔𝑡) 𝑑𝑡 15. ∫1 √𝑡(1 + 𝑡)2 𝑑𝑡
0.5 1 2
16. ∫0.2 cos(3𝑥) 𝑑𝑥 17. ∫0 sin(4𝑡) 𝑑𝑡 18. ∫1 cos(𝜃) + sin(𝜃) 𝑑𝜃
19. The force 𝐹 newtons acting on a body at a distance 𝑥 metres from a fixed point
1
is given by 𝐹 = 3𝑥 + . Given that the work done by the force in moving the
𝑥2
𝑥
body from 𝑥 = 𝑥1 to 𝑥 = 𝑥2 is 𝑊 = ∫𝑥 2 𝐹 𝑑𝑥, find the work done when the
1
body moves from 𝑥 = 1 to 𝑥 = 3.
20. In each of the following, find the total area between the given curve and the 𝑥
axis between the given values of 𝑥. You should check whether or not the curve
crosses the 𝑥 axis between the given values of 𝑥 first.
(a) 𝑦 = 𝑥 2 − 2𝑥 between 𝑥 = 0 and 𝑥 = 2
𝜋 𝜋
(b) 𝑦 = 3 cos(2𝑥) between 𝑥 = and 𝑥 =
6 3
1
(c) 𝑦= between 𝑥 = 0 and 𝑥 = 2
𝑒 2𝑥
21. Find the area between the curve 𝑥𝑦 = 4 and the 𝑥 axis between 𝑥 = 3 and 𝑥 =
6.
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