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Worksheet: Definite Integrals as Limits of Riemann Sums

Q1:

Express l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 π‘₯ 𝑖 𝑖 ο„š 𝑒 2 βˆ’ 4 π‘₯ Ξ” π‘₯ 𝑖 as a definite integral on the interval [ βˆ’ 5 , βˆ’ 3 ] .

  • A ο„Έ 𝑒 2 βˆ’ 4 π‘₯ π‘₯ ∞ 1 π‘₯ d
  • B ο„Έ 𝑒 2 βˆ’ 4 π‘₯ π‘₯ 𝑛 1 π‘₯ d
  • C ο„Έ 𝑒 2 βˆ’ 4 π‘₯ π‘₯ 𝑛 βˆ’ 5 π‘₯ d
  • D ο„Έ 𝑒 2 βˆ’ 4 π‘₯ π‘₯ βˆ’ 3 βˆ’ 5 π‘₯ d
  • E ο„Έ 𝑒 2 βˆ’ 4 π‘₯ π‘₯ ∞ βˆ’ 5 π‘₯ d

Q2:

Express l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 π‘₯ 𝑖 𝑖 ο„š 3 𝑒 5 βˆ’ π‘₯ Ξ” π‘₯ 𝑖 as a definite integral on the interval [ βˆ’ 5 , 1 ] .

  • A ο„Έ 3 𝑒 5 βˆ’ π‘₯ π‘₯ ∞ 1 π‘₯ d
  • B ο„Έ 3 𝑒 5 βˆ’ π‘₯ π‘₯ 𝑛 1 π‘₯ d
  • C ο„Έ 3 𝑒 5 βˆ’ π‘₯ π‘₯ 𝑛 βˆ’ 5 π‘₯ d
  • D ο„Έ 3 𝑒 5 βˆ’ π‘₯ π‘₯ 1 βˆ’ 5 π‘₯ d
  • E ο„Έ 3 𝑒 5 βˆ’ π‘₯ π‘₯ ∞ βˆ’ 5 π‘₯ d

Q3:

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

  • A38
  • B18
  • C36
  • D42
  • E16

Q4:

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

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

Q5:

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

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

Q6:

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†’ ∞ 2 𝑖 = βˆ’ 5 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 βˆ’ 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ 5 + 7 𝑖 𝑛 
  • C 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 
  • D 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 
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 7 𝑛 ο„Ÿ 7 βˆ’ 4 ο€½ 7 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†’ ∞ 3 𝑖 = βˆ’ 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 βˆ’ 4 𝑛 ο„Ÿ 3 ο€½ 1 + 4 𝑖 𝑛  + 2
  • C l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 0 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ βˆ’ 1 + 4 𝑖 𝑛  + 2
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ βˆ’ 1 + 4 𝑖 𝑛  + 2
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 4 𝑛 ο„Ÿ 3 ο€½ 4 𝑖 𝑛  + 2