3.6 Numerical integration  (Page 3/11)

In the two previous examples, we were able to compare our estimate of an integral with the actual value of the integral; however, we do not typically have this luxury. In general, if we are approximating an integral, we are doing so because we cannot compute the exact value of the integral itself easily. Therefore, it is often helpful to be able to determine an upper bound for the error in an approximation of an integral. The following theorem provides error bounds for the midpoint and trapezoidal rules. The theorem is stated without proof.

We can use these bounds to determine the value of $n$ necessary to guarantee that the error in an estimate is less than a specified value.

Simpson’s rule

With the midpoint rule, we estimated areas of regions under curves by using rectangles. In a sense, we approximated the curve with piecewise constant functions. With the trapezoidal rule, we approximated the curve by using piecewise linear functions. What if we were, instead, to approximate a curve using piecewise quadratic functions? With Simpson’s rule    , we do just this. We partition the interval into an even number of subintervals, each of equal width. Over the first pair of subintervals we approximate ${\displaystyle {\int }_{x_0}^{x_2}f\left(x\right)dx}$ with ${\displaystyle {\int }_{x_0}^{x_2}p\left(x\right)dx,}$ where $p\left(x\right)=A{x}^2+Bx+C$ is the quadratic function passing through $(x_0,f\left(x_0\right)),$ $(x_1,f\left(x_1\right)),$ and $(x_2,f\left(x_2\right))$ ( [link] ). Over the next pair of subintervals we approximate ${\displaystyle {\int }_{x_2}^{x_4}f\left(x\right)dx}$ with the integral of another quadratic function passing through $(x_2,f\left(x_2\right)),$ $(x_3,f\left(x_3\right)),$ and $(x_4,f\left(x_4\right)).$ This process is continued with each successive pair of subintervals.