Worksheet: The Trapezoidal Rule

In this worksheet, we will practice using numerical integration and estimating and approximating 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 .

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 16 times as much for intervals as for intervals.
  • C The error is 4 times as much for intervals as for intervals.
  • D The error is 16 times as much for intervals as for intervals.
  • E The error is 4 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 16 times as much for intervals as for intervals.
  • EThe error is 4 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
  • B
  • C0.059
  • D
  • E0.59

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

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

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 the quadrilaterals have the common base 𝐸 𝐹
  • Dbecause 𝐴 𝐡 = 𝐴 𝐡 βˆ— βˆ—

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?

  • A48, an overestimate
  • B28, an underestimate
  • C16, an underestimate
  • D32, an overestimate
  • E28, 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 ) = 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 ) β‰₯ ο„Έ 𝑓 ( π‘₯ ) π‘₯ 𝑏 π‘Ž if 𝑓 is concave up on [ π‘Ž , 𝑏 ] .

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