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Lesson: Definite Integrals as Limits of Riemann Sums

In this Lesson, we will learn how to use Riemann sum to express, evaluate, and estimate the definite integral.

Worksheet • 7 Questions

Q1:

Express ο„Έ 3 π‘₯ π‘₯ 9 3 6 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š 1 8 𝑛 ο€½ 3 + 6 𝑖 𝑛 
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š 1 8 𝑛 ο€½ 6 𝑖 𝑛 
  • C l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š βˆ’ 1 8 𝑛 ο€½ 3 βˆ’ 6 𝑖 𝑛 
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 0 6 ο„š 1 8 𝑛 ο€½ 3 + 6 𝑖 𝑛 
  • E l i m 𝑛 β†’ ∞ 9 𝑖 = 3 6 ο„š 1 8 𝑛 ο€½ 3 + 6 𝑖 𝑛 

Q2:

Without evaluating the limit, express ο„Έ √ 7 βˆ’ 4 π‘₯ π‘₯ 2 βˆ’ 5 2 d as a limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ βˆ’ 5 + 7 𝑖 𝑛 
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ 7 𝑖 𝑛 
  • C l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š βˆ’ 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ 5 + 7 𝑖 𝑛 
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 0 2 ο„š 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ βˆ’ 5 + 7 𝑖 𝑛 
  • E l i m 𝑛 β†’ ∞ 2 𝑖 = βˆ’ 5 2 ο„š 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ βˆ’ 5 + 7 𝑖 𝑛 

Q3:

Express as a definite integral on the interval .

  • A
  • B
  • C
  • D
  • E

Q4:

Evaluate ο„Έ ο€Ή π‘₯ βˆ’ 3 π‘₯  π‘₯ 2 βˆ’ 4 2 d using the limit of Riemann sums.

  • A42
  • B16
  • C18
  • D36
  • E38

Q5:

Express as a definite integral on the interval .

  • A
  • B
  • C
  • D
  • E

Q6:

Express ο„Έ 5 π‘₯ π‘₯ 6 4 6 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š 1 0 𝑛 ο€½ 4 + 2 𝑖 𝑛 
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š 1 0 𝑛 ο€½ 2 𝑖 𝑛 
  • C l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 6 ο„š βˆ’ 1 0 𝑛 ο€½ 4 βˆ’ 2 𝑖 𝑛 
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 0 6 ο„š 1 0 𝑛 ο€½ 4 + 2 𝑖 𝑛 
  • E l i m 𝑛 β†’ ∞ 6 𝑖 = 4 6 ο„š 1 0 𝑛 ο€½ 4 + 2 𝑖 𝑛 

Q7:

Without evaluating the limit, express ο„Έ √ 3 π‘₯ + 2 π‘₯ 3 βˆ’ 1 2 d as a limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ βˆ’ 1 + 4 𝑖 𝑛  + 2
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ 4 𝑖 𝑛  + 2
  • C l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š βˆ’ 4 𝑛 ο„Ÿ 3 ο€½ 1 + 4 𝑖 𝑛  + 2
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 0 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ βˆ’ 1 + 4 𝑖 𝑛  + 2
  • E l i m 𝑛 β†’ ∞ 3 𝑖 = βˆ’ 1 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ βˆ’ 1 + 4 𝑖 𝑛  + 2
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