# 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.

5 | 7 | 9 | 11 | 13 | 15 | 17 | |

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.

1 | 2 | 3 | 4 | 5 | 6 | 7 | |

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 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.

3 | 7 | 11 | 15 | 19 | 23 | 27 | |

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 underestimate
- B48, an overestimate
- C16, 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 .

10 | 13 | 16 | 19 | 22 | 25 | |

1 | 4 | 8 | 10 |

- Alower bound: , upper bound: 66
- Blower bound: , upper bound: 3
- Clower bound: , upper bound: 20
- Dlower bound: , upper bound: 1
- Elower bound: , upper bound: 60

**Q12: **

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

- A48, an underestimate
- B13, an underestimate
- C48, an overestimate
- D16, an underestimate
- E16, 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
- D7.686, 7.769
- E7.769, 7.686