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Worksheet: Using Definite Integral Properties

Q1:

Suppose 𝑓 has absolute minimum value π‘š and absolute maximum value 𝑀 . Given that π‘Ž β©½ π‘₯ β©½ 𝑏 , which property of integrals allows you to decide between what two values ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž d must lie?

  • A π‘Ž β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑏 𝑏 π‘Ž d
  • B π‘š β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 𝑏 π‘Ž d
  • C ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d
  • D π‘š ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ 𝑀 ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d
  • E π‘š ( 𝑏 βˆ’ π‘Ž ) β©½ ο„Έ 𝑓 ( π‘₯ ) π‘₯ β©½ ( 𝑏 βˆ’ π‘Ž ) 𝑏 π‘Ž d

Q2:

If ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 2 5 βˆ’ 4 d and ο„Έ 𝑔 ( π‘₯ ) π‘₯ = 7 4 5 βˆ’ 4 d , find ο„Έ [ 2 𝑓 ( π‘₯ ) βˆ’ 4 𝑔 ( π‘₯ ) ] π‘₯ . 5 βˆ’ 4 d

Q3:

The function is continuous on and satisfies . Determine .

Q4:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 5 1 βˆ’ 2 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 7 1 8 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ 8 βˆ’ 2 d ?

Q5:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 0 βˆ’ 4 βˆ’ 8 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 3 βˆ’ 4 7 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ 7 βˆ’ 8 d ?

Q6:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 6 6 βˆ’ 5 βˆ’ 7 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 7 βˆ’ 5 2 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 βˆ’ 7 d ?

Q7:

If ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 6 βˆ’ 2 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 1 9 βˆ’ 2 d , find ο„Έ 𝑓 ( π‘₯ ) π‘₯ 9 6 d .

Q8:

If ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 . 4 2 βˆ’ 5 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 1 . 4 βˆ’ 1 βˆ’ 5 d , find ο„Έ 𝑓 ( π‘₯ ) π‘₯ 2 βˆ’ 1 d .

Q9:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 1 8 4 βˆ’ 6 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 6 βˆ’ 6 βˆ’ 1 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ 4 βˆ’ 1 d ?

Q10:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 8 6 7 1 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 3 7 1 5 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ 7 5 d ?

Q11:

The function 𝑓 is continuous on ℝ . Given ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 1 βˆ’ 3 βˆ’ 9 d and ο„Έ 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 2 3 βˆ’ 9 βˆ’ 8 d , what is ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ 3 βˆ’ 8 d ?

Q12:

If ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 7 1 βˆ’ 9 d and ο„Έ 𝑔 ( π‘₯ ) π‘₯ = βˆ’ 7 1 βˆ’ 9 d , determine the value of ο„Έ [ 𝑓 ( π‘₯ ) + 𝑔 ( π‘₯ ) ] π‘₯ 1 βˆ’ 9 d .

Q13:

If ο„Έ 𝑔 ( π‘₯ ) π‘₯ = 1 0 8 βˆ’ 7 d , determine the value of ο„Έ 7 𝑔 ( π‘₯ ) π‘₯ βˆ’ 7 8 d .

Q14:

Suppose that on [ βˆ’ 2 , 5 ] , the values of 𝑓 lie in the interval [ π‘š , 𝑀 ] . Between which bounds does ο„Έ 𝑓 ( π‘₯ ) π‘₯ 5 βˆ’ 2 d lie?

  • A βˆ’ 2 ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 5 5 βˆ’ 2 d
  • B π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 𝑀 5 βˆ’ 2 d
  • C 7 ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 𝑀 5 βˆ’ 2 d
  • D 7 π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 𝑀 5 βˆ’ 2 d
  • E 7 π‘š ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ ≀ 7 5 βˆ’ 2 d

Q15:

Write ο„Έ 𝑓 ( π‘₯ ) π‘₯ + ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ ο„Έ 𝑓 ( π‘₯ ) π‘₯ 3 βˆ’ 2 4 3 0 βˆ’ 2 d d d in the form ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž d .

  • A ο„Έ 𝑓 ( π‘₯ ) π‘₯ 3 0 d
  • B ο„Έ 𝑓 ( π‘₯ ) π‘₯ 0 4 d
  • C ο„Έ 𝑓 ( π‘₯ ) π‘₯ 0 3 d
  • D ο„Έ 𝑓 ( π‘₯ ) π‘₯ 4 0 d
  • E ο„Έ 𝑓 ( π‘₯ ) π‘₯ βˆ’ 2 3 d