4-6Trapezoidal Rule: A Mathematics Academy Production

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.

Trapezoid Rule (x o,y o ) (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) (x 4,y 4 ) h

Trapezoidal Rule: where [a,b] is partitioned into n subintervals of equal length h = (b – a)/n This gives us a better approximation than either left or right rectangles. Trapezoid Rule

Actual area under curve:

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

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

Averaging right and left rectangles… …produces the area of a trapezoid:

The area of a trapezoid can be found with the formula:

Notice how all the values in the middle show up twice when summing the areas of each trapezoid. This allows us to come up with a simple general formula seen back in slide 4.