# Worksheet: The Trapezoidal Rule

In this worksheet, we will practice using numerical integration and estimating and approximating definite integrals using the trapezoidal rule.

**Q1: **

The following table shows how the midpoint and tapezium rule perform on estimating . The error is the difference from the actual value of the integral .

Intervals | Mid | Trap | ||
---|---|---|---|---|

2 | 0.790588 | 0.775 | ||

8 | 0.7857236 | 0.7847471 | ||

32 | 0.785418 | 0.785357 | ||

128 | 0.785399 | 0.785396 | 216.3397448 | |

512 | 0.785398 | 16.33974482 | 0.785398 | 16.33974482 |

What appears to be true of the ratio of successive errors for the midpoint rule?

- A The error is 16 times as much for intervals as for intervals.
- B The error is 16 times as much for intervals as for intervals.
- C The error is 4 times as much for intervals as for intervals.
- D The error is 16 times as much for intervals as for intervals.
- E The error is 4 times as much for intervals as for intervals.

What appears to be true of the ratio of successive errors for the trapezium rule?

- AThe error is 16 times as much for intervals as for intervals.
- BThe error is 4 times as much for intervals as for intervals.
- CThe error is 16 times as much for intervals as for intervals.
- DThe error is 16 times as much for intervals as for intervals.
- EThe error is 4 times as much for intervals as for intervals.

The midpoint rule underestimates the integral and the trapezium rule overestimates it. What geometric property of the graph of explains this?

- AThe graph is concave down.
- B The graph is concave up.

For a fixed number of intervals , what appears to be the relation between the midpoint rule and the trapezium rule errors?

- AThe trapezium rule error is times the midpoint error.
- BThe trapezium rule error is times the midpoint error.
- CThe trapezium rule error is times the midpoint error.
- DThe trapezium rule error is times the midpoint error.
- EThe trapezium rule error is 2 times the midpoint error.

Simpsonβs rule can be expressed as the weighted average . Using the table above with , we get a Simpsonβs error of . Using technology, find the actual error to 3 decimal places.

- A
- B
- C0.059
- D
- E0.59

**Q2: **

For a fixed function on a given interval, let be the estimated integral using the trapezoidal rule with subintervals. Use the following diagram to relate the concavity of with how well estimates .

- A if is concave down on .
- B if is concave down on .
- CThere is no conclusion about the comparison between the estimate and actual values.
- D if is concave down on .

**Q3: **

Consider a function whose graph is concave up on the interval .

The line that determines points and is the tangent line to over the midpoint of segment .

Which quadrilateral has the area given by , the trapezoidal rule estimate of the integral ?

- A
- B
- C
- D
- E

Which quadrilateral has the area given by , the midpoint rule estimate of the integral ?

- A
- B
- C
- D
- E

Why is Area Area?

- Abecause the triangles and are congruent
- Bbecause and are equal
- Cbecause the quadrilaterals have the common base
- Dbecause

What relationship can you deduce about the numbers , , and in the case where the graph is concave up?

- A
- B
- C
- D

**Q4: **

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

- A48, an overestimate
- B28, an underestimate
- C16, an underestimate
- D32, an overestimate
- E28, an overestimate

**Q5: **

For a fixed function on a given interval, let be the estimated integral using the trapezoidal rule with subintervals. Use the following diagram to relate the concavity of with how well estimates .

- A if is concave up on .
- B if is concave up on .
- CThere is no conclusion about the comparison between the estimate and actual values.
- D if is concave up on .