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In this lesson, we will learn how to estimate and approximate 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 .

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 trapezium rule?

The midpoint rule underestimates the integral and the trapezium 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 trapezium rule errors?

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

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 .

Q3:

Consider a function whose graph π¦ = π ( π₯ ) is concave up on the interval πΉ πΈ .

The line π΄ π΅ that determines points π΄ and π΅ is the tangent line to π¦ = π ( π₯ ) over the midpoint π of segment πΉ πΈ .

Which quadrilateral has the area given by T r a p ( 1 ) , the trapezoidal rule estimate of the integral οΈ π ( π₯ ) π₯ ο€ ο₯ d ?

Which quadrilateral has the area given by M i d ( 1 ) , the midpoint rule estimate of the integral οΈ π ( π₯ ) π₯ ο€ ο₯ d ?

Why is Area ( π΄ π΅ πΈ πΉ ) = β β Area ( π΄ π΅ πΈ πΉ ) ?

What relationship can you deduce about the numbers M i d ( 1 ) , T r a p ( 1 ) , and οΈ π ( π₯ ) π₯ ο€ ο₯ d in the case where the graph is concave up?

Q4:

Calculate the trapezoidal rule estimate of οΈ π₯ + 2 π₯ 4 0 2 d with π = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

Q5:

For a fixed function on a given interval, let T r a p ( π ) be the estimated integral using the trapezoidal rule with π subintervals. Use the following diagram to relate the concavity of π with how well T r a p ( π ) estimates οΈ π ( π₯ ) π₯ π π d .

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