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
View Answer

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

Answer: b
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
View Answer

Answer: c
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
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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
View Answer

Answer: d
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
View Answer

Answer: c
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.
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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
View Answer

Answer: c
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
b) Adding intervals property
c) Adding integral property
d) Adding integrand property
View Answer

Answer: b
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.
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8. What is the name of the property \(\int_a^b\)f(x)dx=-\(\int_b^a\)f(x)dx?
a) Reverse integral property
b) Adding intervals property
c) Zero interval property
d) Adding integrand property
View Answer

Answer: a
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
b) Adding intervals property
c) Zero-length interval property
d) Adding integrand property
View Answer

Answer: b
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.
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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
b) Adding intervals property
c) Zero-length interval property
d) Adding integrand property
View Answer

Answer: a
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
View Answer

Answer: c
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
View Answer

Answer: c
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
View Answer

Answer: c
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)
View Answer

Answer: b
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
View Answer

Answer: c
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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