Questions and Answers

PART A 

QUESTIONS AND ANSWERS

1. Evaluate interpreting it in terms of area.

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2. Evaluate interpreting in terms of area.

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3. Express in [2, 6], as a definite integral.

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4. Express the integral as limit of Riemann sum.

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5. Evaluate interpreting as area.

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6.

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= 2 × 37 + 3 × 16 = 122

7. If

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8. Express as definite integral

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9. Find the derivative of

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10. Find

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11. If y =

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12. Find the fallacy

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13. Evaluate

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14. Evaluate

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15. If

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16.

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17. Prove that is a constant, find the constant

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18. If In a = 2, ln b = −3 then find

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In ab2 = In a +2 ln b = 2 + 2(-3) = - 4

19. Evaluate

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20. Evaluate

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21. With a suitable substitution transformation the integral to one in the list of integrals?

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22. If y =

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23. If f(x) = 12, then, f' is continuous and What is the value of f(4)?

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24. Evaluate 

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25. Verify by differentiation that the formula is correct

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We have to check it is correct or not by differentiation

Differentiate w.r. to x, we get

= cos x – sin2 x cos x

= cos x (1 - sin2x)

= cos x cos2x = cos3x, which is true

⸫ the given formula is correct

26. Evaluate using a formula in the list (without substitution)

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27. Evaluate

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28. If f is continuous and

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29. If ƒ is continuous on [a, b], then

True or false

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False, we cannot bring variable out of the integral sign.

30. True or false

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31.   represents the area under the curve y = x − x2 True or false

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Since f(x) is not ≥ 0 ∀x ∈ [0, 2], we can not say it is the area under the curve.

32. True or false

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True

33. Evaluate

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34. Evaluate 

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35. Test the convergence of

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So, the integral is not convergent ⇒ it is divergent

36. Test the convergence of

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Hence, the integral is convergent

37. Indicate the type of improper integral and test its convergence

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⸫ The integral is convergent.

38.

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39.   is a proper or improper integral? Why?

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It is improper integral of second kind or type.

40.

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Hence convergent True

41.

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42. Evaluate

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⸪ sin3x cos2x is an odd function

43. Let f be a positive function

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44. Suppose ƒ and g are continuous functions in (—∞, ∞)

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45. Find the value of

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46. Find

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⸫ ƒ is even function

47. If

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48. Find

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49. Evaluate

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50. Evaluate

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51. Evaluate

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52.

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Let t = sin θ – cos θ

⸫ dt = cos θ + sin θ