Lesson: Definite Integral Properties

In this lesson, we will learn how to use definite integration properties, such as the order of integration limits, zero-width limits, sums, and differences.

Worksheet: Definite Integral Properties • 15 Questions

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?

Q2:

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 .

Q3:

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 .

Q4:

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 .

Q5:

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 .

Q6:

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 ?

Q7:

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 ?

Q8:

The function 𝑓 is continuous on [ βˆ’ 4 , 4 ] and satisfies ο„Έ 𝑓 ( π‘₯ ) π‘₯ = 9 4 0 d . Determine ο„Έ [ 𝑓 ( π‘₯ ) βˆ’ 6 ] π‘₯ 4 0 d .

Q9:

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 .

Q10:

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

Q11:

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

Q12:

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 ?

Q13:

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 ?

Q14:

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 ?

Q15:

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 ?

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