Lesson Worksheet: Definite Integrals as Limits of Riemann Sums Mathematics • Higher Education

In this worksheet, we will practice interpreting a definite integral as the limit of a Riemann sum when the size of the partitions tends to zero.

Q1:

Express ο„Έ3π‘₯π‘₯d as the limit of Riemann sums.

  • AlimοŠβ†’βˆžοŠοƒοŠ²οŠ¦οŠ¬ο„š18𝑛3+6𝑖𝑛
  • BlimοŠβ†’βˆžοŠ―οƒοŠ²οŠ©οŠ¬ο„š18𝑛3+6𝑖𝑛
  • ClimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„šβˆ’18𝑛3βˆ’6𝑖𝑛
  • DlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„š18𝑛3+6𝑖𝑛
  • ElimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„š18𝑛6𝑖𝑛

Q2:

Without evaluating the limit, express ο„Έβˆš7βˆ’4π‘₯π‘₯d as a limit of Riemann sums.

  • AlimοŠβ†’βˆžοŠοƒοŠ²οŠ¦οŠ¨ο„š7π‘›ο„Ÿ7βˆ’4ο€½βˆ’5+7𝑖𝑛
  • BlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„š7π‘›ο„Ÿ7βˆ’4ο€½βˆ’5+7𝑖𝑛
  • ClimοŠβ†’βˆžοŠ¨οƒοŠ²οŠ±οŠ«οŠ¨ο„š7π‘›ο„Ÿ7βˆ’4ο€½βˆ’5+7𝑖𝑛
  • DlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„šβˆ’7π‘›ο„Ÿ7βˆ’4ο€½5+7𝑖𝑛
  • ElimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„š7π‘›ο„Ÿ7βˆ’4ο€½7𝑖𝑛

Q3:

Express limοŠβ†’βˆžοŠοƒοŠ²οŠ§ο—οƒοƒο„šπ‘’2βˆ’4π‘₯Ξ”π‘₯ as a definite integral on the interval [βˆ’5,βˆ’3].

  • A𝑒2βˆ’4π‘₯π‘₯οŠοŠ±οŠ«ο—d
  • B𝑒2βˆ’4π‘₯π‘₯βˆžοŠ§ο—d
  • C𝑒2βˆ’4π‘₯π‘₯οŠ±οŠ©οŠ±οŠ«ο—d
  • D𝑒2βˆ’4π‘₯π‘₯οŠοŠ§ο—d
  • E𝑒2βˆ’4π‘₯π‘₯βˆžοŠ±οŠ«ο—d

Q4:

Evaluate ο„Έο€Ήπ‘₯βˆ’3π‘₯π‘₯οŠͺd using the limit of Riemann sums.

Q5:

Express limοŠβ†’βˆžοŠοƒοŠ²οŠ§ο—οƒοƒο„š3𝑒5βˆ’π‘₯Ξ”π‘₯ as a definite integral on the interval [βˆ’5,1].

  • Aο„Έ3𝑒5βˆ’π‘₯π‘₯οŠοŠ±οŠ«ο—d
  • Bο„Έ3𝑒5βˆ’π‘₯π‘₯βˆžοŠ§ο—d
  • Cο„Έ3𝑒5βˆ’π‘₯π‘₯οŠ§οŠ±οŠ«ο—d
  • Dο„Έ3𝑒5βˆ’π‘₯π‘₯οŠοŠ§ο—d
  • Eο„Έ3𝑒5βˆ’π‘₯π‘₯βˆžοŠ±οŠ«ο—d

Q6:

Express ο„Έ5π‘₯π‘₯οŠͺd as the limit of Riemann sums.

  • AlimοŠβ†’βˆžοŠοƒοŠ²οŠ¦οŠ¬ο„š10𝑛4+2𝑖𝑛
  • BlimοŠβ†’βˆžοŠ¬οƒοŠ²οŠͺοŠ¬ο„š10𝑛4+2𝑖𝑛
  • ClimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„šβˆ’10𝑛4βˆ’2𝑖𝑛
  • DlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„š10𝑛4+2𝑖𝑛
  • ElimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¬ο„š10𝑛2𝑖𝑛

Q7:

Without evaluating the limit, express ο„Έβˆš3π‘₯+2π‘₯d as a limit of Riemann sums.

  • AlimοŠβ†’βˆžοŠοƒοŠ²οŠ¦οŠ¨ο„š4π‘›ο„Ÿ3ο€½βˆ’1+4𝑖𝑛+2
  • BlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„š4π‘›ο„Ÿ3ο€½βˆ’1+4𝑖𝑛+2
  • ClimοŠβ†’βˆžοŠ©οƒοŠ²οŠ±οŠ§οŠ¨ο„š4π‘›ο„Ÿ3ο€½βˆ’1+4𝑖𝑛+2
  • DlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„šβˆ’4π‘›ο„Ÿ3ο€½1+4𝑖𝑛+2
  • ElimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„š4π‘›ο„Ÿ3ο€½4𝑖𝑛+2

Q8:

Evaluate ο„Έο€Ή4π‘₯βˆ’4π‘₯π‘₯d by taking the limit of Riemann sums.

Q9:

Using Riemann sums, express limοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ«οŠ¬ο„šβˆ’π‘–π‘› as an integral.

  • Aβˆ’ο„Έπ‘₯π‘₯d
  • Bο„Έπ‘₯π‘₯∞d
  • Cβˆ’ο„Έπ‘₯π‘₯d
  • Dβˆ’ο„Έπ‘₯π‘₯d
  • Eβˆ’ο„Έπ‘₯π‘₯d

Q10:

Express ο„Έ35π‘₯π‘₯οŠ¨οŽ„οŠ¦sind as the limit of Riemann sums.

  • AlimsinοŠβ†’βˆžοŠ¨οŽ„οƒοŠ²οŠ¦ο„šβˆ’6πœ‹π‘›ο€½10πœ‹π‘–π‘›ο‰
  • BlimsinοŠβ†’βˆžοŠοƒοŠ²οŠ§ο„šβˆ’6πœ‹π‘›ο€½10πœ‹π‘–π‘›ο‰
  • ClimsinοŠβ†’βˆžοŠ¨οŽ„οƒοŠ²οŠ§ο„š6πœ‹π‘›ο€½10πœ‹π‘–π‘›ο‰
  • DlimsinοŠβ†’βˆžοŠ¨οŽ„οƒοŠ²οŠ¦ο„š6πœ‹π‘›ο€½10πœ‹π‘–π‘›ο‰
  • ElimsinοŠβ†’βˆžοŠοƒοŠ²οŠ§ο„š6πœ‹π‘›ο€½10πœ‹π‘–π‘›ο‰

Q11:

Evaluate ο„Έο€Ήπ‘₯βˆ’3π‘₯βˆ’5π‘₯οŠͺd using the limit of Riemann sums.

Q12:

Express limοŠβ†’βˆžοŠο—οŠ²οŠ§ο—οŠοŠ¨1π‘›ο„š54βˆ’ο€»ο‡ as a definite integral.

  • Aο„Έ54βˆ’π‘₯π‘₯d
  • Bβˆ’ο„Έ54βˆ’π‘₯π‘₯d
  • Cο„Έ54βˆ’π‘₯π‘₯d
  • Dο„Έ54βˆ’π‘₯π‘₯d
  • Eο„Έ54βˆ’π‘₯π‘₯∞d

Q13:

Evaluate ο„Έο€Ήπ‘₯βˆ’2π‘₯π‘₯d using the limit of Riemann sums.

  • Aβˆ’727
  • Bβˆ’16
  • Cβˆ’53
  • Dβˆ’12
  • E13

Q14:

Express ο„Έο€Ό2π‘₯βˆ’5π‘₯π‘₯d as the limit of Riemann sums.

  • AlimοŠβ†’βˆžοŠ«οƒοŠ²οŠ¨οŠ¨οŠ©οƒοŠο„š3𝑛2ο€½2+3π‘–π‘›ο‰βˆ’5ο€»2+
  • BlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨ο„š3𝑛2ο€½3π‘–π‘›ο‰βˆ’5𝑛3𝑖
  • ClimοŠβ†’βˆžοŠ«οƒοŠ²οŠ¦οŠ¨οŠ©οƒοŠο„š3𝑛2ο€½2+3π‘–π‘›ο‰βˆ’5ο€»2+
  • DlimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨οŠ©οƒοŠο„š3𝑛2ο€½2+3π‘–π‘›ο‰βˆ’5ο€»2+
  • ElimοŠβ†’βˆžοŠοƒοŠ²οŠ§οŠ¨οŠ©οƒοŠο„šβˆ’3𝑛2ο€½2βˆ’3π‘–π‘›ο‰βˆ’5ο€»2βˆ’ο‡ο§

Q15:

Evaluate ο„Έ(βˆ’π‘₯βˆ’4)π‘₯οŠͺd using the limit of Riemann sums.

Q16:

Evaluate ο„Έο€Ήβˆ’π‘₯βˆ’2π‘₯+3π‘₯d using the limit of Riemann sums.

  • Aβˆ’283
  • Bβˆ’23
  • Cβˆ’163
  • D203
  • Eβˆ’323

Q17:

Evaluate ο„Έβˆ’π‘₯π‘₯d using the limit of Riemann sums.

  • Aβˆ’32
  • B20
  • Cβˆ’16
  • Dβˆ’20
  • Eβˆ’26

Q18:

Evaluate ο„Έ(βˆ’π‘₯βˆ’2)π‘₯d using the limit of Riemann sums.

  • Aβˆ’4
  • Bβˆ’8
  • Cβˆ’10
  • Dβˆ’7
  • Eβˆ’6

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