# 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.

Simpsonβs rule can be expressed as the weighted average . Using the table above with , we get a Simpsonβs 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 .