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Lesson: The Trapezoidal Rule

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

Worksheet • 5 Questions

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 .

Intervals Mid Trap
2 0.790588 0.775
8 0.7857236 0.7847471
32 0.785418 0.785357
128 0.785399 0.785396 216.3397448
512 0.785398 16.33974482 0.785398 16.33974482

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

  • A The error is 16 times as much for intervals as for intervals.
  • B The error is 4 times as much for intervals as for intervals.
  • C The error is 16 times as much for intervals as for intervals.
  • D The error is 4 times as much for intervals as for intervals.
  • E The error is 16 times as much for intervals as for intervals.

What appears to be true of the ratio of successive errors for the trapezium 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 4 times as much for intervals as for intervals.
  • EThe error is 16 times as much for intervals as for intervals.

The midpoint rule underestimates the integral and the trapezium rule overestimates it. What geometric property of the graph of explains this?

  • AThe graph is concave down.
  • B The graph is concave up.

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

  • AThe trapezium rule error is times the midpoint error.
  • BThe trapezium rule error is times the midpoint error.
  • CThe trapezium rule error is times the midpoint error.
  • DThe trapezium rule error is times the midpoint error.
  • EThe trapezium rule 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
  • B0.59
  • C0.059
  • D
  • E

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 .
  • B if is concave down on .
  • CThere is no conclusion about the comparison between the estimate and actual values.
  • D if is concave down on .

Q3:

Consider a function whose graph 𝑦 = 𝑓 ( π‘₯ ) is concave up on the interval 𝐹 𝐸 .

The line 𝐴 𝐡 that determines points 𝐴 and 𝐡 is the tangent line to 𝑦 = 𝑓 ( π‘₯ ) over the midpoint 𝑀 of segment 𝐹 𝐸 .

Which quadrilateral has the area given by T r a p ( 1 ) , the trapezoidal rule estimate of the integral ο„Έ 𝑓 ( π‘₯ ) π‘₯  οŒ₯ d ?

  • A 𝐢 𝐷 𝐸 𝐹
  • B 𝐢 𝐹 𝐸 𝐡 βˆ—
  • C 𝐴 𝐡 𝐸 𝐹 βˆ— βˆ—
  • D 𝐢 𝐹 𝐸 𝐡
  • E 𝐴 𝐡 𝐸 𝐹

Which quadrilateral has the area given by M i d ( 1 ) , the midpoint rule estimate of the integral ο„Έ 𝑓 ( π‘₯ ) π‘₯  οŒ₯ d ?

  • A 𝐴 𝐡 𝐸 𝐹 βˆ— βˆ—
  • B 𝐢 𝐹 𝐸 𝐡 βˆ—
  • C 𝐢 𝐷 𝐸 𝐹
  • D 𝐢 𝐹 𝐸 𝐡
  • E 𝐴 𝐡 𝐸 𝐹

Why is Area ( 𝐴 𝐡 𝐸 𝐹 ) = βˆ— βˆ— Area ( 𝐴 𝐡 𝐸 𝐹 ) ?

  • Abecause the triangles 𝑋 𝐴 𝐴 βˆ— and 𝑋 𝐡 𝐡 βˆ— are congruent
  • Bbecause 𝑋 𝐴 𝐴 βˆ— and 𝑋 𝐷 𝐡 βˆ— are equal
  • Cbecause 𝐴 𝐡 = 𝐴 𝐡 βˆ— βˆ—
  • Dbecause the quadrilaterals have the common base 𝐸 𝐹

What relationship can you deduce about the numbers M i d ( 1 ) , T r a p ( 1 ) , and ο„Έ 𝑓 ( π‘₯ ) π‘₯  οŒ₯ d in the case where the graph is concave up?

  • A M i d d T r a p ( 1 ) < ο„Έ 𝑓 ( π‘₯ ) π‘₯ < ( 1 )  οŒ₯
  • B M i d d T r a p ( 1 ) > ο„Έ 𝑓 ( π‘₯ ) π‘₯ > ( 1 )  οŒ₯
  • C M i d d T r a p ( 1 ) > ο„Έ 𝑓 ( π‘₯ ) π‘₯ < ( 1 )  οŒ₯
  • D M i d d T r a p ( 1 ) < ο„Έ 𝑓 ( π‘₯ ) π‘₯ > ( 1 )  οŒ₯

Q4:

Calculate the trapezoidal rule estimate of ο„Έ π‘₯ + 2 π‘₯ 4 0 2 d with 𝑛 = 2 subintervals. Is the result an overestimate or underestimate of the actual value?

  • A32, an overestimate
  • B28, an overestimate
  • C28, an underestimate
  • D16, an underestimate
  • E48, an overestimate

Q5:

For a fixed function on a given interval, let T r a p ( 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 T r a p ( 𝑛 ) estimates ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž d .

  • A T r a p d ( 𝑛 ) β‰₯ ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž if 𝑓 is concave up on [ π‘Ž , 𝑏 ] .
  • B T r a p d ( 𝑛 ) ≀ ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž if 𝑓 is concave up on [ π‘Ž , 𝑏 ] .
  • CThere is no conclusion about the comparison between the estimate and actual values.
  • D T r a p d ( 𝑛 ) = ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž if 𝑓 is concave up on [ π‘Ž , 𝑏 ] .
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