# Worksheet: Numerical Integration: Riemann Sums

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

Q11:

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

• A48, an underestimate
• B18, an underestimate
• C16, an underestimate
• D16, an overestimate
• E48, an overestimate

Q12:

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

• A16, an overestimate
• B13, an underestimate
• C48, an underestimate
• D16, an underestimate
• E48, an overestimate

Q13:

Suppose that and . Find two estimates of to three decimal places using the left and then the right endpoint methods with 8 rectangles.

• A7.843, 7.686
• B0.7689, 0.686
• C7.769, 7.843
• D7.769, 7.686
• E7.686, 7.769

Q14:

Calculate the right endpoint estimate of with 8 subintervals of equal width. Give your answer to 2 decimal places. State whether your estimate is an overestimate or underestimate of the actual value of the integral.

• A2.77, an underestimate
• B2.77, an overestimate
• C4.78, an overestimate
• D3.61, an underestimate
• E4.78, an underestimate

Q15:

Calculate the left endpoint estimate of with 8 subintervals of equal width.