Worksheet: Properties of Definite Integrals

In this worksheet, we will practice using properties of definite integration, such as the order of integration limits, zero-width limits, sums, and differences.

Q1:

Suppose 𝑓 has absolute minimum value 𝑚 and absolute maximum value 𝑀 . Given that 𝑎 𝑥 𝑏 , which property of integrals allows you to decide between what two values 𝑓 ( 𝑥 ) 𝑥 d must lie?

  • A 𝑚 ( 𝑏 𝑎 ) 𝑓 ( 𝑥 ) 𝑥 ( 𝑏 𝑎 ) d
  • B 𝑎 𝑓 ( 𝑥 ) 𝑥 𝑏 d
  • C 𝑚 𝑓 ( 𝑥 ) 𝑥 𝑀 d
  • D 𝑚 ( 𝑏 𝑎 ) 𝑓 ( 𝑥 ) 𝑥 𝑀 ( 𝑏 𝑎 ) d
  • E ( 𝑏 𝑎 ) 𝑓 ( 𝑥 ) 𝑥 𝑀 ( 𝑏 𝑎 ) d

Q2:

Write 𝑓 ( 𝑥 ) 𝑥 + 𝑓 ( 𝑥 ) 𝑥 𝑓 ( 𝑥 ) 𝑥 d d d in the form 𝑓 ( 𝑥 ) 𝑥 d .

  • A 𝑓 ( 𝑥 ) 𝑥 d
  • B 𝑓 ( 𝑥 ) 𝑥 d
  • C 𝑓 ( 𝑥 ) 𝑥 d
  • D 𝑓 ( 𝑥 ) 𝑥 d
  • E 𝑓 ( 𝑥 ) 𝑥 d

Q3:

If 𝑔 ( 𝑥 ) 𝑥 = 1 0 d , determine the value of 7 𝑔 ( 𝑥 ) 𝑥 d .

Q4:

If 𝑓 ( 𝑥 ) 𝑥 = 1 d and 𝑓 ( 𝑥 ) 𝑥 = 1 1 d , find 𝑓 ( 𝑥 ) 𝑥 d .

Q5:

If 𝑓 ( 𝑥 ) 𝑥 = 2 . 4 d and 𝑓 ( 𝑥 ) 𝑥 = 1 . 4 d , find 𝑓 ( 𝑥 ) 𝑥 d .

Q6:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 9 5 d and 𝑓 ( 𝑥 ) 𝑥 = 7 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q7:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 1 8 d and 𝑓 ( 𝑥 ) 𝑥 = 6 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q8:

The function 𝑓 is continuous on [ 4 , 4 ] and satisfies 𝑓 ( 𝑥 ) 𝑥 = 9 d . Determine [ 𝑓 ( 𝑥 ) 6 ] 𝑥 d .

Q9:

If 𝑓 ( 𝑥 ) 𝑥 = 7 d and 𝑔 ( 𝑥 ) 𝑥 = 7 d , determine the value of [ 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) ] 𝑥 d .

Q10:

If 𝑓 ( 𝑥 ) 𝑥 = 8 2 d and 𝑔 ( 𝑥 ) 𝑥 = 7 4 d , find [ 2 𝑓 ( 𝑥 ) 4 𝑔 ( 𝑥 ) ] 𝑥 . d

Q11:

Suppose that on [ 2 , 5 ] , the values of 𝑓 lie in the interval [ 𝑚 , 𝑀 ] . Between which bounds does 𝑓 ( 𝑥 ) 𝑥 d lie?

  • A 2 𝑓 ( 𝑥 ) 𝑥 5 d
  • B 7 𝑚 𝑓 ( 𝑥 ) 𝑥 7 𝑀 d
  • C 7 𝑓 ( 𝑥 ) 𝑥 7 𝑀 d
  • D 7 𝑚 𝑓 ( 𝑥 ) 𝑥 7 d
  • E 𝑚 𝑓 ( 𝑥 ) 𝑥 𝑀 d

Q12:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 8 0 d and 𝑓 ( 𝑥 ) 𝑥 = 3 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q13:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 6 6 d and 𝑓 ( 𝑥 ) 𝑥 = 2 7 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q14:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 8 6 d and 𝑓 ( 𝑥 ) 𝑥 = 3 7 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q15:

The function 𝑓 is continuous on . Given 𝑓 ( 𝑥 ) 𝑥 = 9 1 d and 𝑓 ( 𝑥 ) 𝑥 = 2 3 d , what is 𝑓 ( 𝑥 ) 𝑥 d ?

Q16:

The function 𝑓 is odd, continuous on [ 1 , 7 ] , and satisfies 𝑓 ( 𝑥 ) 𝑥 = 1 7 d . Determine 𝑓 ( 𝑥 ) 𝑥 d .

Q17:

The function 𝑓 is even, continuous on [ 8 , 8 ] , and satisfies 𝑓 ( 𝑥 ) 𝑥 = 1 9 d and 𝑓 ( 𝑥 ) 𝑥 = 1 3 d . Determine 𝑓 ( 𝑥 ) 𝑥 d .

  • A 7 2
  • B 4 5 2
  • C 3 2
  • D 2 1 2

Q18:

If the even function 𝑓 is continuous over the interval [ 4 , 4 ] , where 𝑓 ( 𝑥 ) 𝑥 = 2 d , determine the value of 𝑓 ( 𝑥 ) 𝑥 d .

Q19:

Determine 7 𝑥 5 𝑥 + 5 𝑥 d .

Q20:

Determine 9 𝑥 6 𝑥 2 𝑥 + 9 𝑥 d .

Q21:

Evaluate 𝑥 𝑥 d .

  • A 1 8 8
  • B 1 4 4
  • C 1 8 9
  • D0
  • E 2 8 9

Q22:

Find 𝑘 𝑥 + 𝑘 𝑥 d d , given that 𝐾 is a constant.

  • A 6 𝑘
  • B 2 4
  • C 1 2 𝑘
  • D 1 4 𝑘

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