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 𝑎𝑥𝑏, which property of integrals allows you to decide between what two values 𝑓(𝑥)𝑥d must lie?

  • A𝑚(𝑏𝑎)𝑓(𝑥)𝑥(𝑏𝑎)d
  • B𝑚(𝑏𝑎)𝑓(𝑥)𝑥𝑀(𝑏𝑎)d
  • C(𝑏𝑎)𝑓(𝑥)𝑥𝑀(𝑏𝑎)d
  • D𝑚𝑓(𝑥)𝑥𝑀d
  • E𝑎𝑓(𝑥)𝑥𝑏d

Q2:

Write 𝑓(𝑥)𝑥+𝑓(𝑥)𝑥𝑓(𝑥)𝑥ddd in the form 𝑓(𝑥)𝑥d.

  • A𝑓(𝑥)𝑥d
  • B𝑓(𝑥)𝑥d
  • C𝑓(𝑥)𝑥d
  • D𝑓(𝑥)𝑥d
  • E𝑓(𝑥)𝑥d

Q3:

If 𝑔(𝑥)𝑥=10d, determine the value of 7𝑔(𝑥)𝑥d.

Q4:

If 𝑓(𝑥)𝑥=1d and 𝑓(𝑥)𝑥=11d, find 𝑓(𝑥)𝑥d.

Q5:

If 𝑓(𝑥)𝑥=2.4d and 𝑓(𝑥)𝑥=1.4d, find 𝑓(𝑥)𝑥d.

Q6:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=95d and 𝑓(𝑥)𝑥=7d, what is 𝑓(𝑥)𝑥d?

Q7:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=18d and 𝑓(𝑥)𝑥=6d, what is 𝑓(𝑥)𝑥d?

Q8:

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

Q9:

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

Q10:

If 𝑓(𝑥)𝑥=82d and 𝑔(𝑥)𝑥=74d, find [2𝑓(𝑥)4𝑔(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)𝑥d lie?

  • A7𝑚𝑓(𝑥)𝑥7d
  • B7𝑚𝑓(𝑥)𝑥7𝑀d
  • C𝑚𝑓(𝑥)𝑥𝑀d
  • D2𝑓(𝑥)𝑥5d
  • E7𝑓(𝑥)𝑥7𝑀d

Q12:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=80d and 𝑓(𝑥)𝑥=3d, what is 𝑓(𝑥)𝑥d?

Q13:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=66d and 𝑓(𝑥)𝑥=27d, what is 𝑓(𝑥)𝑥d?

Q14:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=86d and 𝑓(𝑥)𝑥=37d, what is 𝑓(𝑥)𝑥d?

Q15:

The function 𝑓 is continuous on . Given 𝑓(𝑥)𝑥=91d and 𝑓(𝑥)𝑥=23d, what is 𝑓(𝑥)𝑥d?

Q16:

The function 𝑓 is odd, continuous on [1,7], and satisfies 𝑓(𝑥)𝑥=17d. Determine 𝑓(𝑥)𝑥d.

Q17:

The function 𝑓 is even, continuous on [8,8], and satisfies 𝑓(𝑥)𝑥=19d and 𝑓(𝑥)𝑥=13d. Determine 𝑓(𝑥)𝑥d.

  • A452
  • B32
  • C212
  • D72

Q18:

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

Q19:

Determine 7𝑥5𝑥+5𝑥d.

Q20:

Determine 9𝑥6𝑥2𝑥+9𝑥d.

Q21:

Evaluate 𝑥𝑥d.

  • A0
  • B189
  • C289
  • D144
  • E188

Q22:

Find 𝑘𝑥+𝑘𝑥dd, given that 𝑘 is a constant.

  • A24
  • B14𝑘
  • C6𝑘
  • D12𝑘

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