Questions on Riemann Integral with Answers

Practice Questions on Riemann Integral with Answers

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Questions on Riemann Integral with Answers


Q1. Let a be a positive real number. Which of the following Integrals are convergent ?

(a) a0 1/x4 dx
(b) a0 1/√x dx
(c) 40 1/xlogex dx
(d) 50 1/x(logx)2 dx


Q2. Let k be a positive integer. The radius of convergence of the series Σn=0 (n!)kxn/(kn)! is

(a) k
(b) k-k
(c) kk
(d) ∞


Q3. Let {an : n≥1} be a sequence of real numbers such that Σn=1 an is convergent and Σn=1 |an| is divergent. Let R be the radius of  convergence of the power series Σn=1∞ anxn then we can conclude that

(a) 0 < R < 1
(b) R = 1
(c) 1 < R < ∞
(d) R = ∞


Q4. Let f be a monotonically increasing function from [0,1] into [0,1]. Which of the following statements is/are true ?

(a) f must be continuous at all but finitely many points in [0,1]
(b) f must be continuous at all but countably many points in [0,1]
(c) f must be Riemann integrable
(d) f must be Lebesgue integrable


Q5. Let a, p be real numbers and a > 1

(a) If p > 1 then -∞ 1/|x|pa dx < ∞
(b) If p > 1/a then -∞ 1/|x|pa dx < ∞
(c) If p < 1/a  then -∞ 1/|x|pa dx < ∞
(d) For any p ∈ R we have -∞ 1/|x|pa dx = ∞


Q6.  Let L = 01 dx/(1+x8). Then

(a) L < 1
(b) L > 1
(c) L < π/4
(d) L > π/4


Q7. Let f, g and h be bounded functions on the closed interval [a,b], such that f(x)≤g(x)≤h(x) for all x ∈ [a,b]. Let P = {a = ao= a1< a2 < … < an = b} be a partition of [a,b]. We denote by U(f,P) and L(f,P), the upper and lower Riemann sums of f with respect to the partition P and similarly for g and h. Which of the following statements is necessarily true ?

(a) If U(h,P) – U(f,P) < 1 then U(g,P) – L(g,P) < 1
(b) If L(h,P) – L(f,P) < 1 then U(g,P) – L(g,P) < 1
(c) If U(h,P) – L(f,P) < 1 then U(g,P) – L(g,P) < 1
(d) If L(h,P) – U(f,P) < 1 then U(g,P) – L(g,P) < 1


Q8. Let f:R → R be a differentiable function such that f’ is bounded. Given a closed and bounded interval [a,b], and a partition P = {a = ao= a1< a2 < … < an = b} of [a,b], let M(f,P) and m(f,P) denote respectively, the upper Riemann sum and the lower Riemann sum of with respect to P. Then

(a) |M(f,P) – ab f(x)dx| ≤ (b – a) sup{|f(x)| : x∈[a,b]}
(b) |m(f,P) – ab f(x)dx| ≤ (b – a) inf{|f(x)| : x∈[a,b]}
(c) |M(f,P) – ab f(x)dx| ≤ (b – a)2 sup{|f'(x)| : x∈[a,b]}
(d) |m(f,P) – ab f(x)dx| ≤ (b – a)2 imf{|f'(x)| : x∈[a,b]}


Q9. Suppose f is an increasing real valued function on [0,∞] with f(x) > 0 ∀x and let g(x) = 1/x 0xf(u)du ; 0< x < ∞. Then which of the following are true :

(a) g(x) ≤ f(x) for all x ∈ (0,∞)
(b) xg(x) ≤ f(x) for all x ∈ (0,∞)
(c) xg(x) ≥ f(0) x ∈ (0,∞)
(d) yg(y) – xg(x) ≤ (y-x) f(y) for all x < y


Q10. Let f:[0,1] → R be defined by f(x) = xcos[π/(2x)] if x ≠ 0, and f(x) = 0 if x = 0 then

(a) f is continuous on [0,1]
(b) f is of bounded variation on [0,1]
(c) f is differentiable [0,1] and its derivative f’ is bounded on (0,1)
(d) f is Riemann integrable on [0,1]


Q11. Let f:[0,1] → R be continuous such that f(t) ≥ 0 for all t in [0,1]. Define g(x) = 0x f(t) dt then

(a) g is monotone and bounded
(b) g is monotone, but not bounded
(c) g is bounded, but not monotone
(d) g is neither monotone nor bounded


Q12. Let f be a continuous function on [0,-1] with f(0) = 1. Let G(a) = 1/a 0a f(x) dx

(a) lima0 G(a) = 1/2
(b) lima0 G(a) = 1
(c) lima0 G(a) = 0
(d) The Limit lima0 G(a) does not exist


Q13. The improper Integral 04 dx/x(4-x) is

(a) Convergent & value in 6
(b) Divergent & diverges to +∞
(c) Convergent & value is 4
(d) None of these


Q14. Let f(x) = x3 – sinx, x ∈ [-π/2, π/2]. Then f is

(a) Bounded and of bounded variation
(b) Unbounded and of bounded variation
(c) Unbounded but not of bounded variation
(d) Bounded but not of bounded variation


Q15. Match the Following

A. 12 x/√(x-1) dx B. 0π dx/(1+cosx)

(i) Divergent & diverges to +∞ (ii) Convergent & value is 4/3

(iii) Divergent & diverges to -∞ (iv) Convergent & value is 8/3

(a) A-i, B-ii
(b) A-iii, B-iv
(c) A-iv, B-i
(d) A-ii, B-iii


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Answer Key

01.(b), (d)06.(a), (d)11.(b)
02.(c)07.(c)12.(c)
03.(b)08.(a)13.(b)
04.(b), (c), (d)09.(a), (d)14.(c)
05.(d)10.(b)15.(a), (d)

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