MAT187 L07-08 - Numerical Integration, Estimating Errors

#MAT187 PCE (L7), PCE (L8)
Rather than finding absolute errors for an approximation, we can instead find upper and lower bounds for the error.

Methods/rules for approximating an integral

1. Left-endpoint rule
   Summing rectangles that intersect the function on the top-left corners of rectangles.
   Produces $error=0$ only if the function is a flat horizontal line.
   Produces underestimate if increasing on interval, concavity does not matter.
2. Right-endpoint rule
   Summing rectangles that intersect the function on the top-right corners of rectangles.
   Produces $error=0$ only if the function is a flat horizontal line.
   Produces underestimate if decreasing on interval, concavity does not matter.
3. Trapezoid rule
   A combination of right and/or left endpoint rules (whichever yields an underestimate), combined with right triangles whose vertices intersect the function at the endpoints of an interval.
   Produces $error=0$ only if the function is perfect flat line, either horizontal or slanted.
   Produces underestimate if concave down, increasing or decreasing does not matter.
4. Midpoint rule
   A rectangular sum where the midpoint of each rectangle intersects the function.
   Produces $error=0$ only if the function is a flat horizontal line.
   Produces an underestimate if concave up, increasing or decreasing does not matter.

For any integral approximation, the error is bounded by an underestimate and an overestimate, which you can achieve using one of the four integral approximation rules above.

But, there is another! This rule is known as Simpson's Rule. Simpson's rule is a way of approximating an integral by finding the area under the curve of a quadratic that intersects the original function at the lower bound, the upper bound, and the midpoint between the two bounds.

Simpson's Rule

To approximate ${\int }_a^{b}p(x)dx$, we can integrate a quadratic $f(x)$ that fulfills:

• $f(a)=p(a)$
• $f(b)=p(b)$
• $f\left(\frac{b-a}{2}\right)=p\left(\frac{b-a}{2}\right)$
  ...and this integration looks like:
  ${\int }_a^{b}p(x)dx\approx \frac{b-a}{6}(p(a)+4p\left(\frac{a+b}{2}\right)+p(b))$

For a quadratic, Simpson's Rule has zero error. For a function that is not a simple polynomial, it may also have very little error, depending on the behaviour of the function.

Note: Simpson's rule can only have 2n-subintervals and must have 2n+1 points.
The formula above is for 2 subintervals. I don't think Siefken's gonna ask for more subintervals. Prayge.