# Mathematics Questions and Answers – Definite Integral

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Definite Integral”.

1. In $$\int_a^b$$f(y) dy, what is ‘a’ called as?
a) Integration
b) Upper limit
c) Lower limit
d) Limit of an integral

Explanation: In $$\int_a^b$$f(y) dy ‘a’ is the called as lower limit and ‘b’ is called the upper limit of the integral. The fuction f in $$\int_a^b$$f(y) dy is called the integrand. The letter ‘y’ is a dummy symbol and can be replaced by any other symbol.

2. The value of $$\int_0^{\pi }$$sin ⁡y dy is 2.
a) True
b) False

Explanation: $$\int_0^{\pi }$$sin⁡ y dy = [-cos y]y0
= – cos π – (- cos 0)
= -(-1)-(-1)
= 2

3. Compute ∫cos(x)-$$\frac {3}{x4}$$dx.
a) sin(x)+$$\frac {3}{4}$$x-7+c
b) sec(x)+$$\frac {3}{4}$$x-3+c
c) sin(x)+$$\frac {3}{4}$$x-3
d) sin(x)+$$\frac {3}{4}$$x-3+c

Explanation: ∫cos(x)-$$\frac {3}{x4}$$dx = ∫cos(x) dx−∫3(x-4) dx
= sin(x)+$$\frac {3}{4}$$x-3+c

4. What is the value of $$\int_2^3$$cos⁡(x)-$$\frac {3}{x4}$$dx .
a) sin (3) – sin (2)
b) sin (3) – sin (9) – $$\frac {19}{288}$$
c) sin (8) – sin (2) – $$\frac {19}{288}$$
d) sin (3) – sin (2) – $$\frac {19}{288}$$

Explanation: $$\int_2^3$$cos⁡(x)-$$\frac {3}{x4}$$dx = $$\int_2^3$$sin(x) dx + $$\int_2^3 \frac {3}{4}$$x-3 dx
= (sin (3) + $$\frac {3}{4}$$3-3) – (sin (2) + $$\frac {3}{4}$$2-3)
= sin (3) – sin (2) – $$\frac {19}{288}$$

5. Evaluate $$\int_7^9$$cos⁡(x)dx.
a) 8 (-sin 9 – sin 7)
b) 8 (sin 9 + sin 7)
c) 8 (sin 9 – sin 7)
d) 7 (sin 9 – sin 7)

Explanation: $$\int_7^9$$8cos⁡(x)dx = 8 $$\int_7^9$$cos⁡(x)dx
= 8 (cos x)97
= 8 (sin 9 – sin 7)

6. Compute $$\int_2^3 \frac {cos⁡x-sin⁡x}{4}$$dx.
a) $$\frac {1}{4}$$ (sin 2 + cos 3 – sin 3 – cos 2)
b) $$\frac {1}{4}$$ (sin 3 – cos 3 – sin 2 – cos 2)
c) $$\frac {1}{4}$$ (sin 3 + cos 3 – sin 2 – cos 2)
d) $$\frac {1}{4}$$ (sin 3 + cos 3 + sin 2 – cos 2)

Explanation: $$\int_2^3 \frac {cos⁡x-sin⁡x}{4}$$dx = $$\frac {1}{4}$$ [sin x – (- cos x)]32
= $$\frac {1}{4}$$ (sin x + cos x)32
= $$\frac {1}{4}$$ (sin 3 + cos 3) – $$\frac {1}{4}$$ (sin 2 + cos 2)
= $$\frac {1}{4}$$ (sin 3 + cos 3 – sin 2 – cos 2)

7. What is y in $$\int_a^b$$f(y) dy called as?
a) Random variable
b) Dummy symbol
c) Integral
d) Integrand

Explanation: In $$\int_a^b$$f(y) dy ‘a’ is the called as lower limit and ‘b’ is called the upper limit of the integral. The fuction ‘f’ in $$\int_a^b$$f(y) dy is called the integrand. The letter ‘y’ is a dummy symbol and can be replaced by any other symbol.

8. The value of $$\int_1^2$$1y5 dy is_____
a) 10.5
b) 56
c) 9
d) 23

Explanation: $$\int_1^2$$1y5 dy = (y6/6)21
= $$\frac {64}{6} – \frac{1}{6}$$
= 10.5

9. The value of $$\int_1^2$$1y5/5dy is _____
a) 12
b) 2.1
c) 21
d) 11.1

Explanation: $$\int_1^2$$1y5/5dy = $$\frac {1}{5}$$(y6/6)21
= $$\frac {1}{5}(\frac {64}{6} – \frac{1}{6})$$
= 2.1

10. Evaluate $$\int_0^{\pi }$$sin⁡x dx.
a) 2
b) 6
c) 17
d) 3

Explanation: $$\int_0^{\pi }$$sin⁡x dx = [- cos x]x0
= – cos π – (-cos 0)
= -(-1)-(-1)
= 2

11. Evaluate $$\int_2^3$$cosx dx.
a) 38.2
b) sin (9) – sin (4)
c) 89.21
d) sin (3) – sin (2)

Explanation: $$\int_2^3$$cosx dx = (sin x)32
= sin (3) – sin (2)

12. Compute $$\int_2^3$$2ex dx.
a) 2(e9 – e4)
b) 84.32
c) 2(e3 – e2)
d) 83.25

Explanation: $$\int_2^3$$2ex dx = 2(ex)32 dx
= 2(e3 – e2)

13. In $$\int_b^a$$f(x) dx, b called as lower limit and a is called as upper limit.
a) False
b) True

Explanation: In $$\int_a^b$$f(y) dy ‘a’ is the called as lower limit and ‘b’ is called the upper limit of the integral. The fuction ‘f’ in $$\int_a^b$$f(y) dy is called the integrand. The letter ‘y’ is a dummy symbol and can be replaced by any other symbol.

14. Compute $$\int_3^6$$9 ex dx.
a) 30.82
b) 9(e6 – e3)
c) 11.23
d) 81(e6 – e3)

Explanation: $$\int_3^6$$9 ex dx = 9(ex)63 dx
= 9(e6 – e3)

15. Evaluate $$\int_3^7$$sin(t)-2cos(t)dt.
a) cos(7) – 2sin(7) + (cos(3) + 2sin(3)
b) -17
c) 12
d) cos(7) – 2sin(7) – (cos(3) + 2sin(3)

Explanation: $$\int_3^7$$sin(t)-2cos(t)dt = (cos(t)−2sin(t))73
= (cos(7) – 2sin(7)) – (cos(3) – 2sin(3))
= cos(7) – 2sin(7) – (cos(3) + 2sin(3)

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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