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(𝑥)=𝑥+2 on interval [0,2] in sigma notation using right Riemann sums with 𝑛 subintervals.

  • A8𝑛𝑖+2
  • B8𝑛𝑖
  • C4𝑛(2𝑛)𝑖+1
  • D4𝑛(2𝑛)𝑖+1
  • E8𝑛𝑖+2

Q2:

Find the lower Riemann sum approximation for 𝑓(𝑥)=5𝑥 on [1,2], given that 𝑛=4 subintervals.

Q3:

Compute the right Riemann sum for 𝑓(𝑥)=(2𝜋𝑥)cos on 0,12, given that there are four subintervals of equal width.

Q4:

Compute the left Riemann sum for 𝑓(𝑥)=1𝑥+2 on [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(𝑥)=1𝑥(𝑥2) 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(𝑥)=1𝑥2 in the interval [3,5] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A𝑖𝑖𝑛
  • B22𝑖+𝑛
  • C2𝑛1𝑖𝑛
  • D1𝑖𝑛
  • E𝑖𝑖𝑛

Q7:

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

  • A3𝑛9𝑖+6𝑛𝑖+𝑛𝑖
  • B3𝑛9𝑖+6𝑛𝑖+𝑛
  • C3𝑛9𝑖+6𝑛𝑖+𝑛
  • D3𝑛9𝑖
  • E3𝑛9𝑖+6𝑛𝑖+𝑛𝑖

Q8:

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

  • A16𝑛𝑖
  • B8𝑛𝑖
  • C16𝑛𝑖
  • D16𝑛𝑖
  • E16𝑛𝑖

Q9:

Represent the area under the curve of the function 𝑓(𝑥)=𝑥+4 in the interval [2,2] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A64𝑛𝑛+2𝑖2𝑛𝑖
  • B64𝑛𝑛+2𝑖
  • C64𝑛2𝑖2𝑛𝑖
  • D64𝑛𝑛+2𝑖2𝑛𝑖
  • E64𝑛𝑛+2𝑖2𝑖

Q10:

Represent the area under the curve of the function 𝑓(𝑥)=𝑥1 on the interval [0,3] in sigma notation using a right Riemann sum with 𝑛 subintervals.

  • A27𝑛𝑖
  • B3𝑛9𝑖𝑛
  • C3𝑛9𝑖𝑛
  • D3𝑛9𝑖𝑛𝑖
  • E27𝑛𝑖

Q11:

Write the midpoint Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6 on the interval [0,4] with 𝑛 subintervals.

  • A4𝑛2𝑖+6
  • B4𝑛2𝑖+6
  • C4𝑛32𝑖+𝑖+𝑛+6
  • D4𝑛32𝑖+𝑖+𝑛+6
  • E32𝑖+𝑖+𝑛+6

Q12:

Write the midpoint Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=12𝑥5 on the interval [1,3] with 𝑛 subintervals.

  • A2𝑛13+
  • B2𝑛13+𝑖+
  • C2𝑛13+𝑖+
  • D2𝑛13+𝑖+
  • E2𝑛12+𝑖+

Q13:

Write the left Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6 on the interval [0,4] with 𝑛 subintervals.

  • A4𝑛32𝑖𝑛+6
  • B4𝑛2𝑖+6
  • C32𝑖𝑛+6
  • D4𝑛32𝑖𝑛+6
  • E4𝑛2𝑖+6

Q14:

Write the left Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6 on the interval [1,3] with 𝑛 subintervals.

  • A2𝑛6+2𝑖
  • B2𝑛8+4𝑛𝑖
  • C2𝑛2+4𝑛𝑖
  • D2𝑛8+4𝑛𝑖
  • E2𝑛2+4𝑛𝑖

Q15:

Write the midpoint Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6 on the interval [1,3] with 𝑛 subintervals.

  • A2𝑛2+4𝑛12+𝑖
  • B2𝑛8+4𝑛12+𝑖
  • C2𝑛8+4𝑛(1+𝑖)
  • D2𝑛8+2𝑛12+𝑖
  • E2𝑛8+4𝑛𝑖

Q16:

Write the midpoint Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6𝑥+1 on the interval [1,3] with 𝑛 subintervals.

  • A2𝑛20𝑛𝑖+12+8𝑛𝑖+12+9
  • B2𝑛2𝑛𝑖+12+4𝑛𝑖+12+3
  • C2𝑛12𝑛𝑖+12+8𝑛𝑖+12+9
  • D2𝑛12𝑛𝑖+12+8𝑛𝑖+12+9
  • E2𝑛6𝑛𝑖+12+4𝑛𝑖+12+3

Q17:

Write the left Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=2𝑥+6𝑥+1 on the interval [1,3] with 𝑛 subintervals.

  • A2𝑛12𝑛𝑖+8𝑛𝑖+9
  • B2𝑛20𝑛𝑖+8𝑛𝑖+9
  • C2𝑛6𝑛𝑖+8𝑛𝑖+9
  • D2𝑛2𝑛𝑖+4𝑛𝑖+3
  • E2𝑛12𝑛𝑖+8𝑛𝑖+9

Q18:

Write the left Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=12𝑥+2 on the interval [2,5] with 𝑛 subintervals.

  • A3𝑛12+
  • B2𝑛1𝑖2
  • C2𝑛1𝑖+2
  • D2𝑛1𝑖+2
  • E2𝑛1𝑖+2

Q19:

Write the left Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=12𝑥+6 on the interval [0,3] with 𝑛 subintervals.

  • A3𝑛1𝑖+6
  • B3𝑛1𝑖++6
  • C3𝑛1𝑖+6
  • D3𝑛1𝑖+6
  • E3𝑛1𝑖++6

Q20:

Write the midpoint Riemann sum approximation in sigma notation of the area under the curve of the function 𝑓(𝑥)=12𝑥+6 on the interval [0,3] with 𝑛 subintervals.

  • A3𝑛1𝑖++6
  • B3𝑛1𝑖++6
  • C3𝑛1𝑖++6
  • D3𝑛1𝑖++6
  • E3𝑛1𝑖++6

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