# Lesson Worksheet: Numerical Integration: Riemann Sums Mathematics • Higher Education

In this worksheet, we will practice using right, left, and midpoint Riemann sums to numerically approximate definite integrals.

Q1:

The table shows the values of a function obtained from an experiment. Estimate using three equal subintervals with left endpoints.

 𝑥 𝑓(𝑥) 5 7 9 11 13 15 17 −3 −1.7 −0.6 0.4 1.8 2.5 3.1

Q2:

Using the midpoint rule with , round to four decimal places.

Q3:

The table shows the values of a function obtained from an experiment. Estimate using three equal subintervals with midpoints.

 𝑥 𝑓(𝑥) 1 2 3 4 5 6 7 −3.3 −2.1 −1.3 −0.1 0.9 2.1 3.1

Q4:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q5:

Estimate using the midpoint rule with , giving your answer to four decimal places.

Q6:

Using the Midpoint Rule with , give an estimate of . Give your answer to four decimal places.

Q7:

Approximate the integral using a Riemann sum with right endpoints. Take the number of rectangles to be 8.

• A
• B
• C
• D
• E

Q8:

The table gives the values of a function obtained from an experiment. Use them to estimate using three equal subintervals with right endpoints.

 𝑥 𝑓(𝑥) 3 7 11 15 19 23 27 −2.6 −1.4 −0.7 0.8 2.3 3.4 4.8

Q9:

Calculate the midpoint rule estimate of with subintervals. Is the result an overestimate or underestimate of the actual value?

• A16, an overestimate
• B16, an underestimate
• C48, an overestimate
• D28, an underestimate
• E28, an overestimate

Q10:

The table gives sampled values of an increasing function . Use the data to give a lower and upper bound for .

 𝑥 𝑓(𝑥) 10 13 16 19 22 25 −11 −3 1 4 8 10
• Alower bound: , upper bound: 1
• Blower bound: , upper bound: 20
• Clower bound: , upper bound: 3
• Dlower bound: , upper bound: 60
• Elower bound: , upper bound: 66

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