Worksheet: Riemann Sums

In this worksheet, we will practice approximating the area under the curve of a function using right, left, and midpoint Riemann sums.

Q1:

Given that 𝑓 ( 𝑥 ) = 4 𝑥 c o s and that 0 𝑥 𝜋 4 , evaluate, to the nearest six decimal places, the Riemann sum for 𝑓 with six subintervals, taking the sample points to be left endpoints.

Q2:

Let 𝑓 ( 𝑥 ) = 5 4 𝑥 over 1 𝑥 2 . Using four subintervals and taking midpoints as sample points, evaluate the Riemann sum of 𝑓 to six decimal places.

Q3:

Given 𝑓 ( 𝑥 ) = 𝑥 4 and 4 𝑥 2 , evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A7
  • B 1 2
  • C5
  • D 5
  • E 1 2

Q4:

Let 𝑓 ( 𝑥 ) = 3 2 𝑥 over the interval 1 𝑥 5 . Evaluate the Riemann sum of 𝑓 using four subintervals and right endpoint sample points, giving your answer to six decimal places.

Q5:

Given 𝑓 ( 𝑥 ) = 2 𝑥 5 and 6 𝑥 4 , evaluate the Riemann sum for 𝑓 with five subintervals, taking sample points to be right endpoints.

Q6:

Use left Riemann sum to approximate the area under the curve of 𝑓 ( 𝑥 ) = 𝑥 on the interval [ 0 , 1 ] . Use subintervals with 𝑛 = 5 .

  • A 4 5
  • B 2 2 5
  • C 4 2 5
  • D 9 5
  • E 9 2 5

Q7:

Use left end point approximation to approximate the area under the curve of 𝑓 ( 𝑥 ) = 𝑥 on the interval [ 0 , 3 ] ; use subintervals with 𝑛 = 6 .

  • A 5 5 4
  • B 5 5 8
  • C 9 1 4
  • D 9 1 8
  • E 5 5 1 6

Q8:

Use a right Riemann sum to approximate the area under the curve of 𝑓 ( 𝑥 ) = 3 𝑥 in the interval [ 0 , 2 ] . Use subintervals with 𝑛 = 4 . Approximate your answer to three decimal places.

Q9:

Use a right Riemann sum to approximate the area under the curve of 𝑓 ( 𝑥 ) = 𝑥 4 in the interval [ 2 , 4 ] . Use subintervals with 𝑛 = 5 .

  • A 1 0 4 5
  • B 1 6 4 5
  • C 2 0 8 2 5
  • D 3 2 8 2 5
  • E 1 6 4 2 5

Q10:

Use left Riemann sum to approximate the area under the curve of 𝑓 ( 𝑥 ) = 1 𝑥 2 on the interval [ 3 , 5 ] ; use subintervals with 𝑛 = 4 .

  • A 1 9 2 0
  • B 7 7 1 2 0
  • C 7 7 6 0
  • D 1 9 1 0
  • E 7 7 3 0

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