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Lesson Worksheet: Riemann Sums and Sigma Notation Mathematics • Higher Education

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.

  • A8π‘›ο„šπ‘–+2οŠ©οŠοƒοŠ²οŠ§οŠ¨
  • B8π‘›ο„šπ‘–οŠ©οŠοƒοŠ²οŠ§οŠ¨
  • C4π‘›ο„š(2𝑛)𝑖+1οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨οŠ¨
  • D4π‘›ο„š(2𝑛)𝑖+1οŠοƒοŠ²οŠ§οŠ¨οŠ¨
  • E8π‘›ο„šπ‘–+2οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨

Q2:

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

Q3:

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

Q4:

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

Q5:

Compute the right Riemann sum for 𝑓(π‘₯)=(2πœ‹π‘₯)cos on 0,12, given that there are four subintervals of equal width.

Q6:

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.

  • A16π‘›ο„šπ‘–οŠͺοŠοƒοŠ²οŠ§οŠͺ
  • B8π‘›ο„šπ‘–οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ©
  • C16π‘›ο„šπ‘–οŠͺοŠοƒοŠ²οŠ§οŠ©
  • D16π‘›ο„šπ‘–οŠͺοŠοŠ±οŠ§οƒοŠ²οŠ¦οŠͺ
  • E16π‘›ο„šπ‘–οŠͺοŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ©

Q7:

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

  • A3π‘›ο„šο€Ή9𝑖+6𝑛𝑖+π‘›π‘–ο…οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ©οŠ¨οŠ¨
  • B3π‘›ο„šο€Ή9𝑖+6𝑛𝑖+π‘›ο…οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨οŠ¨
  • C3π‘›ο„šο€Ή9𝑖+6𝑛𝑖+π‘›ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨οŠ¨
  • D3π‘›ο„š9π‘–οŠ¨οŠοƒοŠ²οŠ§οŠ¨
  • E3π‘›ο„šο€Ή9𝑖+6𝑛𝑖+π‘›π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ©οŠ¨οŠ¨

Q8:

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.

  • A32π‘›ο„šο€Ήπ‘›+2π‘–βˆ’2π‘›π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨οŠ¨
  • B32π‘›ο„šο€Ήπ‘›+2π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨οŠ¨
  • C32π‘›ο„šο€Ή2π‘–βˆ’2π‘›π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨
  • D32π‘›ο„šο€Ήπ‘›+2π‘–βˆ’2π‘›π‘–ο…οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨οŠ¨
  • E32π‘›ο„šο€Ήπ‘›+2π‘–βˆ’2π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨

Q9:

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

  • AοŠοƒοŠ²οŠ§ο„šπ‘–π‘–βˆ’π‘›
  • BοŠοƒοŠ²οŠ§ο„š22𝑖+𝑛
  • C2π‘›ο„š1π‘–βˆ’π‘›οŠοƒοŠ²οŠ§
  • DοŠοŠ±οŠ§οƒοŠ²οŠ¦ο„š1π‘–βˆ’π‘›
  • EοŠοŠ±οŠ§οƒοŠ²οŠ¦ο„šπ‘–π‘–βˆ’π‘›

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.

  • A27π‘›ο„šπ‘–οŠ©οŠοƒοŠ²οŠ§οŠ¨
  • B3π‘›ο„šο€Ή9π‘–βˆ’π‘›ο…οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨οŠ¨
  • C3π‘›ο„šο€Ή9π‘–βˆ’π‘›ο…οŠ©οŠοƒοŠ²οŠ§οŠ¨οŠ¨
  • D3π‘›ο„šο€Ή9π‘–βˆ’π‘›π‘–ο…οŠ©οŠοƒοŠ²οŠ§οŠ©οŠ¨
  • E27π‘›ο„šπ‘–οŠ©οŠοŠ±οŠ§οƒοŠ²οŠ¦οŠ¨

This lesson includes 12 additional questions and 90 additional question variations for subscribers.

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