TRB PG Assistant - Riemann integral - fundamental theorem of calculus One Line Questions & Answers

1. The Riemann integral of a constant function f(x) = c over [a, b] is: — c * (b - a)

2. Let f(x) = x^2 on [-2, 2]. What is the integral of f(x) from -2 to 2? — 32/3

3. The integral of a function f from a to a is always: — 0

4. Consider the function f(x) = 1/x on [1, 2]. What is its integral? — ln(2)

5. What is the integral of f(x) = cos(x) from 0 to pi/2? — 1

6. Let f(x) = x^3. What is the definite integral of f(x) from -1 to 1? — 0

7. What is the integral of f(x) = x from 1 to 3? — 8

8. What is the integral of f(x) = 2x + 1 from 0 to 2? — 6

9. Consider the function f(x) = x^2. Using FTC2, what is the definite integral of f(x) from 0 to 2? — 8/3

10. If f is Riemann integrable on [a, b], then integral(f(x) dx from a to b) = - integral(f(x) dx from b to a). This property is called: — Reversal of interval

11. The property that if f is Riemann integrable on [a, b] and m <= f(x) <= M for all x in [a, b], then m(b-a) <= integral(f(x) dx from a to b) <= M(b-a) is known as: — Integral inequality property

12. If f is Riemann integrable on [a, b], and c is a constant, then the integral of c*f(x) from a to b is: — c * integral(f(x) dx from a to b)

13. The Darboux integral is equivalent to the Riemann integral for functions that are: — Riemann integrable

14. Let G(x) = integral(sin(t) dt from 0 to x). According to FTC1, what is G'(x)? — sin(x)

15. What is the integral of f(x) = e^x from 0 to 1? — e - 1

16. Let G(x) = integral(e^(t^2) dt from 0 to x). By FTC1, G'(x) is: — e^(x^2)

17. If f is Riemann integrable on [a, b], then the integral of |f(x)| from a to b is: — Less than or equal to the absolute value of the integral of f(x) from a to b.

18. Which of the following is a consequence of FTC1? — Every continuous function has an antiderivative.

19. What is the primary condition for a function f to be Riemann integrable on an interval [a, b]? — f must be bounded on [a, b] and have a set of discontinuities of measure zero.

20. If f is Riemann integrable on [a, b] and F'(x) = f(x) for all x in (a, b), and F is continuous on [a, b], then FTC2 requires: — f to be continuous on [a, b].

21. If f is continuous on [a, b] and G(x) = integral(f(t) dt from a to x), then what is the value of G(a)? — 0

22. If F(x) is an antiderivative of f(x), then the integral of f(x) from b to a is: — F(a) - F(b)

23. If f is continuous on [a, b], and F is any antiderivative of f, then the integral of f(x) from a to b is given by: — F(b) - F(a)

24. If F'(x) = f(x) for all x in [a, b], and f is Riemann integrable on [a, b], then FTC2 implies that the integral of f(x) from a to b is: — F(b) - F(a)

25. Which of the following functions is NOT Riemann integrable on [0, 1]? — f(x) = 1/x

26. The statement 'If f is continuous on [a, b], then f is Riemann integrable on [a, b]' is: — True

27. The condition for a function to be Riemann integrable is that it must be bounded and its set of discontinuities must have: — measure zero

28. The set of discontinuities of a Riemann integrable function on [a, b] must have: — zero measure

29. What does the second Fundamental Theorem of Calculus (FTC2) state? — If f is Riemann integrable on [a, b] and F is an antiderivative of f on [a, b], then the integral of f(x) from a to b is F(b) - F(a).

30. What does the first Fundamental Theorem of Calculus (FTC1) state? — If f is continuous on [a, b], then the function G(x) = integral(f(t) dt from a to x) is differentiable on (a, b) and G'(x) = f(x).

31. The Riemann integral is defined using: — limits of Riemann sums.

32. If f is Riemann integrable on [a, b] and f(x) >= 0 for all x in [a, b], then: — integral(f(x) dx from a to b) >= 0

33. If f is Riemann integrable on [a, b] and M is the maximum value of f on [a, b], then: — integral(f(x) dx from a to b) <= M * (b - a)

34. If f is Riemann integrable on [a, b], and f(x) = 0 for all x in [a, b] except for a finite number of points, then: — integral(f(x) dx from a to b) = 0

35. If f is continuous on [a, b] and F'(x) = f(x) for all x in [a, b], then FTC2 implies: — integral(f(x) dx from a to b) = F(b) - F(a)

36. If f is Riemann integrable on [a, b] and c is a point such that a < c < b, then which property holds? — integral(f(x) dx from a to b) = integral(f(x) dx from a to c) + integral(f(x) dx from c to b)

37. If f and g are Riemann integrable on [a, b] and f(x) <= g(x) for all x in [a, b], then: — integral(f(x) dx from a to b) <= integral(g(x) dx from a to b)

38. FTC1 is crucial for understanding the relationship between differentiation and integration as: — Differentiation is the inverse of integration.

39. The integral of f(x) = tan(x) from 0 to pi/4 is: — ln(sqrt(2))

40. Let f(x) = |x|. Is f(x) Riemann integrable on [-1, 1]? — Yes, because it is continuous on [-1, 1].

41. Let G(x) = integral(sqrt(1+t^2) dt from 0 to x). What is G'(x)? — sqrt(1+x^2)

42. If f is continuous on [a, b], then the function G(x) = integral(f(t) dt from a to x) represents: — An antiderivative of f.

43. The integral of a function f over [a, b] is defined as the limit of Riemann sums if: — The partition is refined indefinitely (mesh tends to zero).

44. Let f(x) = x^2. Which statement is true regarding its integral on [0, 1]? — The integral is 1/3, and f is continuous.

45. If f(x) = c (a constant) on [a, b], then integral(f(x) dx from a to b) represents: — The area of a rectangle with width (b-a) and height c.

46. The Fundamental Theorem of Calculus connects the concept of the definite integral to: — The derivative of a function.

47. Let G(x) = integral(ln(t) dt from 1 to x). What is G'(x)? — ln(x)

48. Let F(x) = integral(t^2 + 1 dt from 2 to x). What is F'(x)? — x^2 + 1

49. Let G(x) = integral(t^3 dt from 1 to x). By FTC1, G'(x) is: — x^3

50. Let f(x) = 1 for x rational and f(x) = 0 for x irrational in [0, 1]. Is f(x) Riemann integrable on [0, 1]? — No, because it is not continuous anywhere.

Riemann integral - fundamental theorem of calculus - One Line Questions