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 .
What appears to be true of the ratio of successive errors for the midpoint rule?
What appears to be true of the ratio of successive errors for the trapezoidal rule?
The midpoint rule underestimates the integral and the trapezoidal rule overestimates it. What geometric property of the graph of explains this?
For a fixed number of intervals , 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 . Using the table above with , we get a Simpson’s error of . Using technology, find the actual error to 3 decimal places.
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 .
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 ?
Which quadrilateral has the area given by , the midpoint rule estimate of the integral ?
Why is Area Area?
What relationship can you deduce about the numbers , , and in the case where the graph is concave up?