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)=4𝑥cos and that 0𝑥𝜋4, 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)=54𝑥 over 1𝑥2. Using four subintervals and taking midpoints as sample points, evaluate the Riemann sum of 𝑓 to six decimal places.

Q3:

Given 𝑓(𝑥)=𝑥4 and 4𝑥2, evaluate the Riemann sum for 𝑓 with six subintervals, taking sample points to be midpoints.

  • A5
  • B12
  • C5
  • D12
  • E7

Q4:

Let 𝑓(𝑥)=32𝑥 over the interval 1𝑥5. 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)=2𝑥5 and 6𝑥4, 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.

  • A45
  • B95
  • C925
  • D425
  • E225

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.

  • A558
  • B914
  • C918
  • D554
  • E5516

Q8:

Use a right Riemann sum to approximate the area under the curve of 𝑓(𝑥)=3𝑥 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)=𝑥4 in the interval [2,4]. Use subintervals with 𝑛=5.

  • A1645
  • B20825
  • C16425
  • D1045
  • E32825

Q10:

Use left Riemann sum to approximate the area under the curve of 𝑓(𝑥)=1𝑥2 on the interval [3,5]; use subintervals with 𝑛=4.

  • A7760
  • B1920
  • C7730
  • D77120
  • E1910

Q11:

Given that 𝑓(𝑥)=𝑥3𝑥+4, with 0𝑥5, calculate the right Riemann sum for 𝑓 with 5 subintervals of equal width.

Q12:

Given that 𝑓(𝑥)=2𝑒, with 3𝑥8, calculate the midpoint Riemann sum for 𝑓 with 5 subintervals of equal width. Give your answer to three decimal places.

  • A60
  • B56.877
  • C71.001
  • D66.082
  • E48.954

Q13:

Use a right Riemann sum with 10 subintervals to estimate the area under the curve 𝑓(𝑥)=1𝑥+3 on the interval [4,6]. Give your answer to four decimal places.

  • A1.3837
  • B0.2482
  • C0.2767
  • D0.2513
  • E0.2545

Q14:

Use a midpoint Riemann sum with 6 subintervals to estimate the area under the curve 𝑓(𝑥)=𝑥4𝑥+2𝑥+6 on the interval [0,3]. Give your answer to two decimal places.

Q15:

Given that 𝑓(𝑥)=(2𝑥)sin, where 0𝑥𝜋2, calculate the left Riemann sum for 𝑓 with 10 subintervals of equal width. Give your answer to five decimal places.

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