# Mathematics Questions and Answers – Properties of Definite Integrals

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

1. What is the difference property of definite integrals?
a) $$\int_a^b$$[-f(x)-g(x)dx
b) $$\int_a^b$$[f(-x)+g(x)dx
c) $$\int_a^b$$[f(x)-g(x)dx
d) $$\int_a^b$$[f(x)+g(x)dx

Explanation: The sum difference property of definite integrals is $$\int_a^b$$[f(x)-g(x)dx
$$\int_a^b$$[f(x)-g(x)dx = $$\int_a^b$$f(x)dx-$$\int_a^b$$g(x)dx

2. The sum property of definite integrals is $$\int_a^b$$[f(x)+g(x)dx?
a) False
b) True

Explanation: The sum property of definite integrals is $$\int_a^b$$[f(x)+g(x)dx
$$\int_a^b$$[f(x)+g(x)dx = $$\int_a^b$$f(x)dx+$$\int_a^b$$g(x)dx
Hence, it is true.

3. What is the constant multiple property of definite integrals?
a) $$\int_a^b$$k⋅f(x)dy
b) $$\int_a^b$$[f(-x)+g(x)dx
c) $$\int_a^b$$k⋅f(x)dx
d) $$\int_a^b$$[f(x)+g(x)dx

Explanation: The constant multiple property of definite integrals is $$\int_a^b$$k⋅f(x)dx
$$\int_a^b$$k⋅f(x)dx = k $$\int_a^b$$f(x)dx

4. What is the reverse integral property of definite integrals?
a) –$$\int_a^b$$f(x)dx=-$$\int_b^a$$g(x)dx
b) –$$\int_a^b$$f(x)dx=-$$\int_b^a$$g(x)dx
c) $$\int_a^b$$f(x)dx=$$\int_b^a$$g(x)dx
d) $$\int_a^b$$f(x)dx=-$$\int_b^a$$f(x)dx

Explanation: In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is $$\int_a^b$$f(x)dx=-$$\int_b^a$$f(x)dx.

5. Identify the zero-length interval property.
a) $$\int_a^b$$f(x)dx = -1
b) $$\int_a^b$$f(x)dx = 1
c) $$\int_a^b$$f(x)dx = 0
d) $$\int_a^b$$f(x)dx = 0.1

Explanation: The zero-length interval property is one of the properties used in definite integrals and they are always positive. The zero-length interval property is $$\int_a^b$$f(x)dx = 0.

6. What is adding intervals property?
a) $$\int_a^c$$f(x)dx+$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx
b) $$\int_a^b$$f(x)dx+$$\int_b^a$$f(x)dx = $$\int_a^c$$f(x) dx
c) $$\int_a^b$$f(x)dx+$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx
d) $$\int_a^b$$f(x)dx-$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx

Explanation: The adding intervals property of definite integrals is $$\int_a^b$$f(x)dx+$$\int_b^c$$f(x)dx.
$$\int_a^b$$f(x)dx+$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx

7. What is the name of the property of $$\int_a^b$$f(x)dx+$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx?
a) Zero interval property

Explanation: $$\int_a^b$$f(x)dx+$$\int_b^c$$(x)dx = $$\int_a^c$$f(x) dx is a property of definite integrals. $$\int_a^b$$f(x)dx+$$\int_b^c$$f(x)dx = $$\int_a^c$$f(x) dx is called as adding intervals property used to combine a lower limit and upper limit of two different integrals.

8. What is the name of the property $$\int_a^b$$f(x)dx=-$$\int_b^a$$f(x)dx?
a) Reverse integral property
c) Zero interval property

Explanation: In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is $$\int_a^b$$f(x)dx=-$$\int_b^a$$f(x)dx.

9. What is the name of the property $$\int_a^b$$f(x)dx = 0?
a) Reverse integral property
c) Zero-length interval property

Explanation: The zero-length interval property is one of the properties used in definite integrals and they are always positive. The zero-length interval property is $$\int_a^b$$f(x)dx = 0.

10. What property this does this equation come under $$\int^1_{-1}$$sin⁡x dx=-$$\int_1^{-1}$$sin⁡x dx?
a) Reverse integral property
c) Zero-length interval property

Explanation: $$\int^1_{-1}$$sin⁡x dx=-$$\int_1^{-1}$$sin⁡x dx comes under the reverse integral property.
In the reverse integral property the upper limits and lower limits are interchanged. The reverse integral property of definite integrals is $$\int_a^b$$f(x)dx=-$$\int_b^a$$f(x)dx.

11. Evaluate $$\int_2^3$$3f(x)-g(x)dx, if $$\int_2^3$$f(x) = 4 and $$\int_2^3$$g(x)dx = 4.
a) 38
b) 12
c) 8
d) 7

Explanation: $$\int_2^3$$3f(x)-g(x)dx = 3 $$\int_2^3$$f(x) – $$\int_2^3$$g(x)dx
= 3(4) – 4
= 8

12. Compute $$\int_3^2$$f(x) dx if $$\int_2^3$$f(x) = 4.
a) – 4
b) 84
c) 2
d) – 8

Explanation: $$\int_3^2$$f(x)dx = – $$\int_2^3$$f(x)dx
= – 4

13. Compute $$\int_8^2$$2f(x)dx if $$\int_2^8$$f(x) = – 3.
a) – 4
b) 84
c) 2
d) – 8

Explanation: $$\int_8^2$$2f(x)dx = -2 $$\int_2^8$$f(x)dx
= – 2(-3)
= 6

14. Compute $$\int_2^6$$7ex dx.
a) 30.82
b) 7(e6 – e2)
c) 11.23
d) 81(e6 – e3)

Explanation: $$\int_2^6$$7ex dx = 7(ex)62 dx
= 7(e6 – e2)

15. Evaluate $$\int_3^7$$2f(x)-g(x)dx, if $$\int_3^7$$f(x) = 4 and $$\int_3^7$$g(x)dx = 2.
a) 38
b) 12
c) 6
d) 7

Explanation: $$\int_3^7$$2f(x)-g(x)dx = 2 $$\int_3^7$$f(x) – $$\int_3^7$$g(x)dx
= 2(4) – 2
= 6

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

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