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

In this lesson, we will learn how to approximate definite integrals using the trapezoidal rule and estimate the error when using it.

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The following table shows how the midpoint and trapezoidal rule perform on estimating ๏„ธ11+๐‘ฅ๐‘ฅ๏Šง๏Šฆ๏Šจd. The error is the difference from the actual value of the integral ๐œ‹4.

What appears to be true of the ratio of successive errors ErrErr(๐‘›)(4๐‘›) for the midpoint rule?

What appears to be true of the ratio of successive errors ErrErr(๐‘›)(4๐‘›) for the trapezoidal rule?

The midpoint rule underestimates the integral and the trapezoidal rule overestimates it. What geometric property of the graph of ๐‘“(๐‘ฅ)=11+๐‘ฅ๏Šช explains this?

For a fixed number of intervals ๐‘›, what appears to be the relation between the midpoint rule and the trapezoidal rule errors?

Simpsonโ€™s rule can be expressed as the weighted average 2+3MidTrap. Using the table above with ๐‘›=8, we get a Simpsonโ€™s error of 0.03ร—10๏Šฑ๏Šฎ. Using technology, find the actual error to 3 decimal places.


For a fixed function on a given interval, let Trap(๐‘›) be the estimated integral using the trapezoidal rule with ๐‘› subintervals. Use the following diagram to relate the concavity of ๐‘“ with how well Trap(๐‘›) estimates ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œป๏Œบd.


Consider a function whose graph ๐‘ฆ=๐‘“(๐‘ฅ) is concave up on the interval ๐น๐ธ.

The line ๐ด๐ต that determines points ๐ด and ๐ต is the tangent line to ๐‘ฆ=๐‘“(๐‘ฅ) over the midpoint ๐‘€ of segment ๐น๐ธ.

Which quadrilateral has the area given by Trap(1), the trapezoidal rule estimate of the integral ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œค๏Œฅd?

Which quadrilateral has the area given by Mid(1), the midpoint rule estimate of the integral ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œค๏Œฅd?

Why is Area(๐ด๐ต๐ธ๐น)=โˆ—โˆ— Area(๐ด๐ต๐ธ๐น)?

What relationship can you deduce about the numbers Mid(1), Trap(1), and ๏„ธ๐‘“(๐‘ฅ)๐‘ฅ๏Œค๏Œฅd in the case where the graph is concave up?

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