Numerical Integration using Trapezoidal Rule or Simpson’s Rule

Approximating Areas with Trapezoidal Rule or Simpson’s Rule 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.

Actual area under curve:

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

Right-hand rectangular approximation (n =4): Approximate area: (too high)

A more accurate approximation can be obtained if the subintervals are treated as trapezoids instead of rectangles. Averaging right and left rectangles gives us trapezoids:

Trapezoidal Rule: ( h = width of subinterval = ) This rule can be used only if h is constant; otherwise, use trapezoid area formula to find area of each subinterval and then sum all subintervals. This gives us a better approximation than either left or right rectangles. Note: This rule and the trapezoid area formula must be memorized for the AP exam.

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

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.

The widths in meters of a kidney-shaped pool were taken at 2-m intervals. Use the Trapezoidal Rule to estimate the area of the pool. p