Strips, chords and honest areas

Mathematics · Calculus & Analysis · ages 17-18

Name ______________________ Date ____________

Estimate the integral of f(x) = x from 0 to 4 with 2 strips by the trapezium rule.
Answer: ______________
The trapezium rule with n strips uses how many y-values?
- n
- n + 1
- 2n
- n - 1
The trapezium rule with n strips uses how many y-values?
- n + 1
- n
- 2n
For a straight line graph, the trapezium rule gives the exact integral.
Circle one: True False
Estimate the integral of x cubed from 0 to 2 with 2 strips by the trapezium rule.
Answer: ______________
Halving the strip width roughly quarters the trapezium rule error for smooth functions.
Circle one: True False
Estimate the integral of x squared from 0 to 4 with 4 strips by the trapezium rule.
- 22
- 64/3
- 16
- 28
Halving the strip width roughly quarters the trapezium rule error for smooth functions. True or false?
Circle one: True False
Estimate the integral of the constant function 5 from 1 to 6 with 5 strips by the trapezium rule.
Answer: ______________
Estimate the integral of 2x + 1 from 0 to 3 with 3 strips by the trapezium rule.
- 13.5
- 9
- 24
- 12

Answer key

For grown-ups. Fold this page away before handing over the rest.

8 · 8. With h = 2 the estimate is (2/2)(0 + 2(2) + 4) = 8, exact for a line.
n + 1 · n + 1. The n strips share n + 1 boundary ordinates including both ends.
n + 1 · The n strips share n + 1 boundary heights including both ends.
True · Each trapezium coincides exactly with the area under a straight segment.
5 · 5. The estimate is (1/2)(0 + 2(1) + 8) = 5, above the exact 4.
True · The error behaves like h squared, so halving h divides the error by about 4.
22 · 22. Halving the strip sum gives (1/2)(0 + 2(1 + 4 + 9) + 16) = 22, above the exact 64/3.
True · True. The error behaves like h squared, so halving h divides the error by about 4.
25 · Every height is 5, giving width 5 times height 5.
12 · 12. With h = 1 the estimate is (1/2)(1 + 2(3 + 5) + 7) = 12, exact for a line.