Lesson: Numerical Integration: The Trapezoidal Rule AP Calculus BC AP Calculus AB

Mathematics

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

Video

15:37

Sample Question Videos

  • 02:59
  • 03:26
  • 05:19

Worksheet: 15 Questions • 6 Videos

Q1:

The following table shows how the midpoint and trapezoidal rule perform on estimating 11+𝑥𝑥d. The error is the difference from the actual value of the integral 𝜋4.

IntervalsMidErr×10TrapErr×10
20.790588518,983.66030.7751,039,816.34
80.785723632,543.660260.784747165,106.33974
320.7854181,983.6602550.7853574,116.339745
1280.78539983.660255170.785396216.3397448
5120.78539816.339744820.78539816.33974482

What appears to be true of the ratio of successive errors ErrErr(𝑛)(4𝑛) for the midpoint rule?

What appears to be true of the ratio of successive errors ErrErr(𝑛)(4𝑛) for the trapezoidal rule?

The midpoint rule underestimates the integral and the trapezoidal rule overestimates it. What geometric property of the graph of 𝑓(𝑥)=11+𝑥 explains this?

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

Simpson’s rule can be expressed as the weighted average 2+3MidTrap. Using the table above with 𝑛=8, we get a Simpson’s error of 0.03×10. Using technology, find the actual error to 3 decimal places.

Q2:

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

Q3:

Consider a function whose graph 𝑦=𝑓(𝑥) is concave up on the interval 𝐹𝐸.

The line 𝐴𝐵 that determines points 𝐴 and 𝐵 is the tangent line to 𝑦=𝑓(𝑥) over the midpoint 𝑀 of segment 𝐹𝐸.

Which quadrilateral has the area given by Trap(1), the trapezoidal rule estimate of the integral 𝑓(𝑥)𝑥d?

Which quadrilateral has the area given by Mid(1), the midpoint rule estimate of the integral 𝑓(𝑥)𝑥d?

Why is Area(𝐴𝐵𝐸𝐹)= Area(𝐴𝐵𝐸𝐹)?

What relationship can you deduce about the numbers Mid(1), Trap(1), and 𝑓(𝑥)𝑥d in the case where the graph is concave up?

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