Worksheet: Definite Integral Properties

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In this worksheet, we will practice using definite integration properties, 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 𝑏 𝑎 d must lie?

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

Q2:

Write 𝑓 ( 𝑥 ) 𝑥 + 𝑓 ( 𝑥 ) 𝑥 − 𝑓 ( 𝑥 ) 𝑥 3 − 2 4 3 0 − 2 d d d in the form 𝑓 ( 𝑥 ) 𝑥 𝑏 𝑎 d .

  • A 𝑓 ( 𝑥 ) 𝑥 3 0 d
  • B 𝑓 ( 𝑥 ) 𝑥 0 4 d
  • C 𝑓 ( 𝑥 ) 𝑥 0 3 d
  • D 𝑓 ( 𝑥 ) 𝑥 4 0 d
  • E 𝑓 ( 𝑥 ) 𝑥 − 2 3 d

Q3:

If 𝑔 ( 𝑥 ) 𝑥 = 1 0 8 − 7 d , determine the value of 7 𝑔 ( 𝑥 ) 𝑥 − 7 8 d .

Q4:

If 𝑓 ( 𝑥 ) 𝑥 = 1 6 − 2 d and 𝑓 ( 𝑥 ) 𝑥 = 1 1 9 − 2 d , find 𝑓 ( 𝑥 ) 𝑥 9 6 d .

Q5:

If 𝑓 ( 𝑥 ) 𝑥 = − 2 . 4 2 − 5 d and 𝑓 ( 𝑥 ) 𝑥 = − 1 . 4 − 1 − 5 d , find 𝑓 ( 𝑥 ) 𝑥 2 − 1 d .

Q6:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 9 5 1 − 2 d and 𝑓 ( 𝑥 ) 𝑥 = 7 1 8 d , what is 𝑓 ( 𝑥 ) 𝑥 8 − 2 d ?

Q7:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 1 8 4 − 6 d and 𝑓 ( 𝑥 ) 𝑥 = 6 − 6 − 1 d , what is 𝑓 ( 𝑥 ) 𝑥 4 − 1 d ?

Q8:

The function 𝑓 is continuous on [ − 4 , 4 ] and satisfies 𝑓 ( 𝑥 ) 𝑥 = 9 4 0 d . Determine [ 𝑓 ( 𝑥 ) − 6 ] 𝑥 4 0 d .

Q9:

If 𝑓 ( 𝑥 ) 𝑥 = 7 1 − 9 d and 𝑔 ( 𝑥 ) 𝑥 = − 7 1 − 9 d , determine the value of [ 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) ] 𝑥 1 − 9 d .

Q10:

If 𝑓 ( 𝑥 ) 𝑥 = 8 2 5 − 4 d and 𝑔 ( 𝑥 ) 𝑥 = 7 4 5 − 4 d , find [ 2 𝑓 ( 𝑥 ) − 4 𝑔 ( 𝑥 ) ] 𝑥 . 5 − 4 d

Q11:

Suppose that on , the values of lie in the interval . Between which bounds does lie?

  • A
  • B
  • C
  • D
  • E

Q12:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 8 0 − 4 − 8 d and 𝑓 ( 𝑥 ) 𝑥 = 3 − 4 7 d , what is 𝑓 ( 𝑥 ) 𝑥 7 − 8 d ?

Q13:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 6 6 − 5 − 7 d and 𝑓 ( 𝑥 ) 𝑥 = − 2 7 − 5 2 d , what is 𝑓 ( 𝑥 ) 𝑥 2 − 7 d ?

Q14:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 8 6 7 1 d and 𝑓 ( 𝑥 ) 𝑥 = 3 7 1 5 d , what is 𝑓 ( 𝑥 ) 𝑥 7 5 d ?

Q15:

The function 𝑓 is continuous on ℝ . Given 𝑓 ( 𝑥 ) 𝑥 = 9 1 − 3 − 9 d and 𝑓 ( 𝑥 ) 𝑥 = − 2 3 − 9 − 8 d , what is 𝑓 ( 𝑥 ) 𝑥 − 3 − 8 d ?

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