5.5 Trapezoidal Rule, p. 306

1. AP Calculus AB/BC 5.5 Trapezoidal Rule, p. 306

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. Averaging right and left rectangles gives us trapezoids:

8. (still too high)

9. Trapezoidal Rule: h = width of subinterval of equal length h = (b – a)/n This gives us a better approximation than either left or right rectangles.

10. Use the Trapezoidal Rule with n = 4 to approximate the value of the integral. Use concavity of the function to predict whether the approximation is an over- or an underestimate. Finally, find the integral’s exact value to check your answer.

11. Now, let’s take a look at the graph of Now, evaluate Notice the graph opens down, or the graph is concave down. Concave down gives an underestimate, and concave up gives an overestimate using the Trapezoidal Rule.

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:

15. Simpson’s Rule: ( h = width of subinterval, n must be even ) Example:

17. Simpson’s rule can also be interpreted as fitting parabolas to sections of the curve, which is why this example came out exactly. Simpson’s rule will usually give a very good approximation with relatively few subintervals. It is especially useful when we have no equation and the data points are determined experimentally. p