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Worksheet: Limit of a Riemann Sum

Q1:

Evaluate ο„Έ ο€Ή 4 π‘₯ βˆ’ 4 π‘₯  π‘₯ 1 0 3 d by taking the limit of Riemann sums.

Q2:

Evaluate ο„Έ ο€Ή π‘₯ βˆ’ 5 π‘₯  π‘₯ 2 0 3 d by taking the limit of Riemann sums.

Q3:

Using Riemann sums, express l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 5 6 ο„š βˆ’ 𝑖 𝑛 as an integral.

  • A βˆ’ ο„Έ π‘₯ π‘₯ 𝑛 0 6 d
  • B βˆ’ ο„Έ π‘₯ π‘₯ 𝑛 0 5 d
  • C ο„Έ π‘₯ π‘₯ ∞ 0 5 d
  • D βˆ’ ο„Έ π‘₯ π‘₯ 1 0 5 d
  • E βˆ’ ο„Έ π‘₯ π‘₯ 1 0 6 d

Q4:

Express ο„Έ 3 5 π‘₯ π‘₯ 2 πœ‹ 0 s i n d as the limit of Riemann sums.

  • A l i m s i n 𝑛 β†’ ∞ 2 πœ‹ 𝑖 = 0 ο„š 6 πœ‹ 𝑛 ο€½ 1 0 πœ‹ 𝑖 𝑛 
  • B l i m s i n 𝑛 β†’ ∞ 𝑛 𝑖 = 1 ο„š βˆ’ 6 πœ‹ 𝑛 ο€½ 1 0 πœ‹ 𝑖 𝑛 
  • C l i m s i n 𝑛 β†’ ∞ 2 πœ‹ 𝑖 = 0 ο„š βˆ’ 6 πœ‹ 𝑛 ο€½ 1 0 πœ‹ 𝑖 𝑛 
  • D l i m s i n 𝑛 β†’ ∞ 𝑛 𝑖 = 1 ο„š 6 πœ‹ 𝑛 ο€½ 1 0 πœ‹ 𝑖 𝑛 
  • E l i m s i n 𝑛 β†’ ∞ 2 πœ‹ 𝑖 = 1 ο„š 6 πœ‹ 𝑛 ο€½ 1 0 πœ‹ 𝑖 𝑛 

Q5:

Evaluate ο„Έ ο€Ή π‘₯ βˆ’ 3 π‘₯ βˆ’ 5  π‘₯ 2 βˆ’ 4 2 d using the limit of Riemann sums.

Q6:

Express l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 𝑖 𝑛 2 1 𝑛 ο„š 5 4 βˆ’ ο€»  as a definite integral.

  • A βˆ’ ο„Έ 5 4 βˆ’ π‘₯ π‘₯ 𝑛 0 2 d
  • B ο„Έ 5 4 βˆ’ π‘₯ π‘₯ 𝑛 0 2 d
  • C ο„Έ 5 4 βˆ’ π‘₯ π‘₯ ∞ 0 2 d
  • D ο„Έ 5 4 βˆ’ π‘₯ π‘₯ 1 0 2 d
  • E ο„Έ 5 4 βˆ’ π‘₯ π‘₯ 0 1 2 d

Q7:

Express l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 𝑖 𝑛 2 1 𝑛 ο„š 1 3 + 5 ο€»  as a definite integral.

  • A βˆ’ ο„Έ 1 3 + 5 π‘₯ π‘₯ 𝑛 0 2 d
  • B ο„Έ 1 3 + 5 π‘₯ π‘₯ 𝑛 0 2 d
  • C ο„Έ 1 3 + 5 π‘₯ π‘₯ ∞ 0 2 d
  • D ο„Έ 1 3 + 5 π‘₯ π‘₯ 1 0 2 d
  • E ο„Έ 1 3 + 5 π‘₯ π‘₯ 0 1 2 d

Q8:

Evaluate ο„Έ ο€Ή π‘₯ βˆ’ 2 π‘₯  π‘₯ 1 0 2 3 d using the limit of Riemann sums.

  • A βˆ’ 5 3
  • B 1 3
  • C βˆ’ 1 2
  • D βˆ’ 1 6
  • E βˆ’ 7 2 7

Q9:

Evaluate ο„Έ ο€Ή 5 π‘₯ βˆ’ 5 π‘₯  π‘₯ 1 0 2 3 d using the limit of Riemann sums.

  • A βˆ’ 1 0 3
  • B 5 3
  • C βˆ’ 5 4
  • D 5 1 2
  • E 1 0 2 7

Q10:

Express ο„Έ ο€Ό 2 π‘₯ βˆ’ 5 π‘₯  π‘₯ 5 2 2 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 5 𝑖 = 2 2 3 𝑖 𝑛 ο„š 3 𝑛  2 ο€½ 2 + 3 𝑖 𝑛  βˆ’ 5 ο€» 2 +  
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 3 𝑖 𝑛 ο„š βˆ’ 3 𝑛  2 ο€½ 2 βˆ’ 3 𝑖 𝑛  βˆ’ 5 ο€» 2 βˆ’  
  • C l i m 𝑛 β†’ ∞ 5 𝑖 = 0 2 3 𝑖 𝑛 ο„š 3 𝑛  2 ο€½ 2 + 3 𝑖 𝑛  βˆ’ 5 ο€» 2 +  
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 3 𝑖 𝑛 ο„š 3 𝑛  2 ο€½ 2 + 3 𝑖 𝑛  βˆ’ 5 ο€» 2 +  
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 3 𝑛  2 ο€½ 3 𝑖 𝑛  βˆ’ 5 𝑛 3 𝑖 

Q11:

Express ο„Έ ο€Ό 4 π‘₯ + 4 π‘₯  π‘₯ βˆ’ 1 βˆ’ 8 2 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ βˆ’ 1 𝑖 = βˆ’ 8 2 7 𝑖 𝑛 ο„š 7 𝑛  4 ο€½ βˆ’ 8 + 7 𝑖 𝑛  + 4 ο€» βˆ’ 8 +  
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 7 𝑖 𝑛 ο„š βˆ’ 7 𝑛  4 ο€½ βˆ’ 8 βˆ’ 7 𝑖 𝑛  + 4 ο€» βˆ’ 8 βˆ’  
  • C l i m 𝑛 β†’ ∞ βˆ’ 1 𝑖 = 0 2 7 𝑖 𝑛 ο„š 7 𝑛  4 ο€½ βˆ’ 8 + 7 𝑖 𝑛  + 4 ο€» βˆ’ 8 +  
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 7 𝑖 𝑛 ο„š 7 𝑛  4 ο€½ βˆ’ 8 + 7 𝑖 𝑛  + 4 ο€» βˆ’ 8 +  
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 7 𝑛  4 ο€½ 7 𝑖 𝑛  + 4 𝑛 7 𝑖 

Q12:

Express ο„Έ ο€Ό 5 π‘₯ + 4 π‘₯  π‘₯ 5 4 2 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 5 𝑖 = 4 2 𝑖 𝑛 ο„š 1 𝑛  5 ο€½ 4 + 𝑖 𝑛  + 4 ο€» 4 +  
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 𝑖 𝑛 ο„š βˆ’ 1 𝑛  5 ο€½ 4 βˆ’ 𝑖 𝑛  + 4 ο€» 4 βˆ’  
  • C l i m 𝑛 β†’ ∞ 5 𝑖 = 0 2 𝑖 𝑛 ο„š 1 𝑛  5 ο€½ 4 + 𝑖 𝑛  + 4 ο€» 4 +  
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 𝑖 𝑛 ο„š 1 𝑛  5 ο€½ 4 + 𝑖 𝑛  + 4 ο€» 4 +  
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 1 𝑛  5 ο€½ 𝑖 𝑛  + 4 𝑛 𝑖 

Q13:

Express ο„Έ ο€Ό 2 π‘₯ + 2 π‘₯  π‘₯ βˆ’ 2 βˆ’ 6 2 d as the limit of Riemann sums.

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

Q14:

Express ο„Έ ο€Ό 4 π‘₯ + 4 π‘₯  π‘₯ 6 1 2 d as the limit of Riemann sums.

  • A l i m 𝑛 β†’ ∞ 6 𝑖 = 1 2 5 𝑖 𝑛 ο„š 5 𝑛  4 ο€½ 1 + 5 𝑖 𝑛  + 4 ο€» 1 +  
  • B l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 5 𝑖 𝑛 ο„š βˆ’ 5 𝑛  4 ο€½ 1 βˆ’ 5 𝑖 𝑛  + 4 ο€» 1 βˆ’  
  • C l i m 𝑛 β†’ ∞ 6 𝑖 = 0 2 5 𝑖 𝑛 ο„š 5 𝑛  4 ο€½ 1 + 5 𝑖 𝑛  + 4 ο€» 1 +  
  • D l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 5 𝑖 𝑛 ο„š 5 𝑛  4 ο€½ 1 + 5 𝑖 𝑛  + 4 ο€» 1 +  
  • E l i m 𝑛 β†’ ∞ 𝑛 𝑖 = 1 2 ο„š 5 𝑛  4 ο€½ 5 𝑖 𝑛  + 4 𝑛 5 𝑖 

Q15:

Evaluate ο„Έ ( βˆ’ π‘₯ βˆ’ 4 ) π‘₯ 2 βˆ’ 4 d using the limit of Riemann sums.

Q16:

Evaluate ο„Έ ( βˆ’ π‘₯ βˆ’ 4 ) π‘₯ 6 βˆ’ 4 d using the limit of Riemann sums.