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Chapter 8: Applications of Integrals

NCERT Solutions for CBSE Class 10 Mathematics — 39 solved questions with detailed explanations.

39

Questions

0

Topics

Solved Questions

Q1. The area bounded by y = x², x = 0, x = 2, and x-axis is:

4/3
8/3
2
4

Difficulty: Easy-Medium · Topic: Area Under Curve

Solution

∫₀² x² dx = [x³/3]₀² = 8/3.

Q2. Area between y=f(x), x-axis from a to b is:

f(b)-f(a)
∫f(x)dx always
[f(x)]²/2
∫|f(x)|dx from a to b

Difficulty: Easy-Medium · Topic: Applications of Integrals

Solution

∫|f(x)|dx from a to b.

Q3. Area between curves y=f(x) and y=g(x) is:

∫(f+g)dx
f×g
∫|f(x)-g(x)|dx
∫f(x)g(x)dx

Difficulty: Easy-Medium · Topic: Applications of Integrals

Solution

∫|f(x)-g(x)|dx.

Q4. Area under y=x^1 from x=0 to x=1:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 1 = 1^2/2 = 0.5.

Q5. Area under y=x^1 from x=0 to x=2:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 2 = 2^2/2 = 2.0.

Q6. Area under y=x^1 from x=0 to x=3:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 3 = 3^2/2 = 4.5.

Q7. Area under y=x^1 from x=0 to x=4:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 4 = 4^2/2 = 8.0.

Q8. Area under y=x^1 from x=0 to x=5:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 5 = 5^2/2 = 12.5.

Q9. Area under y=x^1 from x=0 to x=6:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 6 = 6^2/2 = 18.0.

Q10. Area under y=x^1 from x=0 to x=7:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^1 dx from 0 to 7 = 7^2/2 = 24.5.

Q11. Area under y=x^2 from x=0 to x=1:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 1 = 1^3/3 = 0.3333.

Q12. Area under y=x^2 from x=0 to x=2:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 2 = 2^3/3 = 2.6667.

Q13. Area under y=x^2 from x=0 to x=3:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 3 = 3^3/3 = 9.0.

Q14. Area under y=x^2 from x=0 to x=4:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 4 = 4^3/3 = 21.3333.

Q15. Area under y=x^2 from x=0 to x=5:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 5 = 5^3/3 = 41.6667.

Q16. Area under y=x^2 from x=0 to x=6:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 6 = 6^3/3 = 72.0.

Q17. Area under y=x^2 from x=0 to x=7:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^2 dx from 0 to 7 = 7^3/3 = 114.3333.

Q18. Area under y=x^3 from x=0 to x=1:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 1 = 1^4/4 = 0.25.

Q19. Area under y=x^3 from x=0 to x=2:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 2 = 2^4/4 = 4.0.

Q20. Area under y=x^3 from x=0 to x=3:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 3 = 3^4/4 = 20.25.

Q21. Area under y=x^3 from x=0 to x=4:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 4 = 4^4/4 = 64.0.

Q22. Area under y=x^3 from x=0 to x=5:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 5 = 5^4/4 = 156.25.

Q23. Area under y=x^3 from x=0 to x=6:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 6 = 6^4/4 = 324.0.

Q24. Area under y=x^3 from x=0 to x=7:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^3 dx from 0 to 7 = 7^4/4 = 600.25.

Q25. Area under y=x^4 from x=0 to x=1:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 1 = 1^5/5 = 0.2.

Q26. Area under y=x^4 from x=0 to x=2:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 2 = 2^5/5 = 6.4.

Q27. Area under y=x^4 from x=0 to x=3:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 3 = 3^5/5 = 48.6.

Q28. Area under y=x^4 from x=0 to x=4:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 4 = 4^5/5 = 204.8.

Q29. Area under y=x^4 from x=0 to x=5:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 5 = 5^5/5 = 625.0.

Q30. Area under y=x^4 from x=0 to x=6:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 6 = 6^5/5 = 1555.2.

Q31. Area under y=x^4 from x=0 to x=7:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^4 dx from 0 to 7 = 7^5/5 = 3361.4.

Q32. Area under y=x^5 from x=0 to x=1:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 1 = 1^6/6 = 0.1667.

Q33. Area under y=x^5 from x=0 to x=2:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 2 = 2^6/6 = 10.6667.

Q34. Area under y=x^5 from x=0 to x=3:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 3 = 3^6/6 = 121.5.

Q35. Area under y=x^5 from x=0 to x=4:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 4 = 4^6/6 = 682.6667.

Q36. Area under y=x^5 from x=0 to x=5:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 5 = 5^6/6 = 2604.1667.

Q37. Area under y=x^5 from x=0 to x=6:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 6 = 6^6/6 = 7776.0.

Q38. Area under y=x^5 from x=0 to x=7:

Difficulty: Easy-Medium · Topic: Area

Solution

∫ x^5 dx from 0 to 7 = 7^6/6 = 19608.1667.

Q39. The area enclosed by y = |x| and y = 2 is:

2
4
8
1

Difficulty: Medium · Topic: Area Between Curves

Solution

y=|x| meets y=2 at x=±2. Area = ∫₋₂² (2-|x|)dx = 2∫₀²(2-x)dx = 2[2x-x²/2]₀² = 2(4-2) = 4.

Other Chapters in Mathematics

Ch 1: Relations & Functions Ch 2: Inverse Trig Functions Ch 3: Matrices Ch 4: Determinants Ch 5: Continuity & Differentiability Ch 6: Applications of Derivatives Ch 7: Integrals Ch 9: Differential Equations Ch 10: Vector Algebra Ch 11: Three Dimensional Geometry Ch 12: Linear Programming Ch 13: Probability

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