Worksheet: Riemann Sums and Sigma Notation

In this worksheet, we will practice using sigma notation with Riemann sums to find the area under a curve.

Q1:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = π‘₯ + 2  on interval [ 0 , 2 ] in sigma notation using right Riemann sums with 𝑛 subintervals.

  • A 4 𝑛 ο„š ( 2 𝑛 ) 𝑖 + 1        
  • B 4 𝑛 ο„š ( 2 𝑛 ) 𝑖 + 1      
  • C 8 𝑛 ο„š 𝑖 + 2        
  • D 8 𝑛 ο„š 𝑖      
  • E 8 𝑛 ο„š 𝑖 + 2      

Q2:

Find the lower Riemann sum approximation for 𝑓 ( π‘₯ ) = 5 βˆ’ π‘₯  on [ 1 , 2 ] , given that 𝑛 = 4 subintervals.

Q3:

Compute the right Riemann sum for 𝑓 ( π‘₯ ) = ( 2 πœ‹ π‘₯ ) c o s on  0 , 1 2  , given that there are four subintervals of equal width.

Q4:

Compute the left Riemann sum for 𝑓 ( π‘₯ ) = 1 π‘₯ + 2  on [ βˆ’ 3 , 3 ] , given that there are six subintervals of equal width. Approximate your answer to nearest two decimal places.

Q5:

Compute the right Riemann sum for 𝑓 ( π‘₯ ) = 1 π‘₯ ( π‘₯ βˆ’ 2 ) on [ 3 , 5 ] , given that there are four subintervals of equal width. Approximate your answer to the nearest three decimal places.

Q6:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = 1 π‘₯ βˆ’ 2 in the interval [ 3 , 5 ] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A       ο„š 𝑖 𝑖 βˆ’ 𝑛
  • B       ο„š 1 𝑖 βˆ’ 𝑛
  • C 2 𝑛 ο„š 1 𝑖 βˆ’ 𝑛    
  • D     ο„š 𝑖 𝑖 βˆ’ 𝑛
  • E     ο„š 2 2 𝑖 + 𝑛

Q7:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = π‘₯ + 2 π‘₯ + 1  on the interval [ 0 , 3 ] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A 3 𝑛 ο„š ο€Ή 9 𝑖 + 6 𝑛 𝑖 + 𝑛 𝑖         
  • B 3 𝑛 ο„š ο€Ή 9 𝑖 + 6 𝑛 𝑖 + 𝑛          
  • C 3 𝑛 ο„š 9 𝑖      
  • D 3 𝑛 ο„š ο€Ή 9 𝑖 + 6 𝑛 𝑖 + 𝑛 𝑖           
  • E 3 𝑛 ο„š ο€Ή 9 𝑖 + 6 𝑛 𝑖 + 𝑛        

Q8:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = π‘₯  on the interval [ 0 , 2 ] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A 1 6 𝑛 ο„š 𝑖 οŠͺ     
  • B 1 6 𝑛 ο„š 𝑖 οŠͺ       οŠͺ
  • C 1 6 𝑛 ο„š 𝑖 οŠͺ     οŠͺ
  • D 1 6 𝑛 ο„š 𝑖 οŠͺ       
  • E 8 𝑛 ο„š 𝑖        

Q9:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = π‘₯ + 4  in the interval [ βˆ’ 2 , 2 ] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A 6 4 𝑛 ο„š ο€Ή 𝑛 + 2 𝑖        
  • B 6 4 𝑛 ο„š ο€Ή 𝑛 + 2 𝑖 βˆ’ 2 𝑛 𝑖        
  • C 6 4 𝑛 ο„š ο€Ή 𝑛 + 2 𝑖 βˆ’ 2 𝑖       
  • D 6 4 𝑛 ο„š ο€Ή 2 𝑖 βˆ’ 2 𝑛 𝑖       
  • E 6 4 𝑛 ο„š ο€Ή 𝑛 + 2 𝑖 βˆ’ 2 𝑛 𝑖          

Q10:

Represent the area under the curve of the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 1  on the interval [ 0 , 3 ] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A 3 𝑛 ο„š ο€Ή 9 𝑖 βˆ’ 𝑛          
  • B 3 𝑛 ο„š ο€Ή 9 𝑖 βˆ’ 𝑛 𝑖        
  • C 3 𝑛 ο„š ο€Ή 9 𝑖 βˆ’ 𝑛        
  • D 2 7 𝑛 ο„š 𝑖        
  • E 2 7 𝑛 ο„š 𝑖      

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