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4.6 Numerical Integration -Trapezoidal Rule -Simpson’s Rule

If we have to evaluate a definite integral involving a function whose antiderivative cannot be found, the Fundamental Theorem of Calculus cannot be used. Thus we must rely on an approximation method.

Trapezoidal Method After creating a bunch of trapezoids, the area for the ith trapezoid is Which implies that Notice that the coefficients of successive terms follow the pattern of …2 2 1

Use the trapezoidal rule to evaluate the definite integral of f(x)=sin(x) when n=4, n=8
And this is the value that we substitute into f(x)..starting with zero, and increasing each time Note that this method is useful when an antiderivative cannot be found. There is an antiderivative for this function so we could have used other methods to find the exact area.

Approximate the area under n(x) = 3x + 3 on the interval [5,6] by using the Trapezoidal Rule for n(x) using 6 subintervals. 19.5

Approximate the area under y(x) = 4x2 + 5x + 5
on the interval [4,5] by using the Trapezoidal Rule for y(x) using 4 subintervals. 108.88

Approximate the area under g(x) = 4x2 + 3x + 3 on the interval [0,4] by using the Trapezoidal Rule for g(x) using 7 subintervals.

Simpson’s Rule Simpson’s Rule uses parabolas instead of rectangles or trapezoids and then sums all of those areas up. So it is a better approximation.

Approximate the area under t(x) = 2x3 + 4x2 + 1 on the interval [1,5] by using the Simpson's Rule for t(x) using 8 subintervals. 481.33

Approximate the area under m(x) = 2x3 + 5x2 + 3x + 2 on the interval [2,6]
by using the Simpson's Rule for m(x) using 6 subintervals.

Notice that the coefficients follow the pattern 1 4 2 4 2 4 …. 4 2 4 1
Let f be continuous on [a,b]. Simpson’s Rule for approximating n has to be even Notice that the coefficients follow the pattern …

Errors in the Trapezoidal and Simpson Rules
We can decide how close of an approximation we want with both of these rules. For example we might want to have an error in approximation less than There is then a formula, that we will learn, that you input the error that you want to be within and it tells you then how many subintervals “n” that you need to be within that measurement of error.

Trapezoid Error Simpson’s Error
If f has a continuous 2nd derivative on [a,b], then the error E in approximating the integral of f(x) by the Trapezoidal Rule is… Simpson’s Error If f has a continuous 4th derivative on [a,b], then the error E in approximating the integral of f(x) by Simpson’s Rule is…

We are not actually going to work out examples because of time limitations but this formula will tell you how many trapezoids or parabolas you need to use in order to be as close to the actual value as you want to be. The max|f’’(x)| value is determined by critical points that were learned in Ch.3.

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