Trapezoidal Rule

1. Trapezoidal Rule Mt. Shasta, California Photo by Vickie Kelly, 1998

2. Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative.

3. Actual area under curve:

4. Approximate area: Left-hand rectangular approximation: (too low)

5. Approximate area: Right-hand rectangular approximation: (too high)

6. Averaging the two: (too high) 1.25% error

7. . The formula we used in geometry to find the area of a trapezoid is:

8. The area under the curve f1(x) is a trapezoid. The integral . NOTE: We used only one trapezoid in this example.

9. Trapezoidal Approximation Averaging right and left rectangles gives us trapezoids:

10. Area of each trapezoid based on (still too high)

11. Trapezoidal Rule: ( h = width of subinterval ) This gives us a better approximation than either left or right rectangles.

12. Approximate area: Compare this with the Midpoint Rule: 0.625% error (too low) The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

13. Trapezoidal Rule: (too high) 1.25% error Midpoint Rule: 0.625% error (too low) Ahhh! Oooh! Wow! Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer!

14. twice midpoint trapezoidal This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules. Midpoint: Trapezoidal: