# Lesson Worksheet: Numerical Integration: The Trapezoidal Rule Mathematics • Higher Education

In this worksheet, we will practice approximating definite integrals using the trapezoidal rule and estimate the error when using it.

Q1:

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

IntervalsMidTrap
20.7905880.7751,039,816.34
80.78572360.784747165,106.33974
320.7854180.7853574,116.339745
1280.7853990.785396216.3397448
5120.78539816.339744820.78539816.33974482

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

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

What appears to be true of the ratio of successive errors for the trapezoidal 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 trapezoidal rule overestimates it. What geometric property of the graph of explains this?

• AThe graph is concave up.
• BThe graph is concave down.

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

• AThe trapezoidal error is times the midpoint error.
• BThe trapezoidal error is times the midpoint error.
• CThe trapezoidal error is times the midpoint error.
• DThe trapezoidal error is times the midpoint error.
• EThe trapezoidal error is 2 times the midpoint error.

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

• A
• B0.059
• C
• 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 .
• BThere is no conclusion about the comparison between the estimate and actual values.
• C if is concave down on .
• 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 quadrilaterals have the common base
• Bbecause and are equal
• Cbecause
• Dbecause the triangles and are congruent

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?

• A28, an overestimate
• B48, an overestimate
• C16, an underestimate
• D28, an underestimate
• E32, 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 .
• BThere is no conclusion about the comparison between the estimate and actual values.
• C if is concave up on .
• D if is concave up on .

Q6:

Use the trapezoidal rule to estimate using five subintervals. Round your answer to three decimal places.

Q7:

Use the trapezoidal rule to estimate using four subintervals.

• A
• B
• C
• D
• E

Q8:

Estimate using the trapezoidal rule with four subintervals. Approximate your answer to two decimal places.

Q9:

Estimate using the trapezoidal rule with four subintervals. Approximate your answer to two decimal places.

Q10:

Use the trapezoidal rule to estimate using four subintervals. Round your answer to three decimal places.

This lesson includes 6 additional questions and 45 additional question variations for subscribers.

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