1. Numerical Approximations of Definite Integrals Mika Seppälä

2. Mika Seppälä: Numerical Integration Riemann Sums The definite integral of a positive function f over an interval [a,b] has been defined by Riemann sums which approximate the area under the graph of f. Taking more division points in the Riemann sums, the approximation of the area of the domain under the graph of f becomes better.

3. Mika Seppälä: Numerical Integration Definite Integrals This definition assumes that the limit does not depend on the various choices in the definition of the Riemann sums.

4. Mika Seppälä: Numerical Integration Numerical Approximations of Definite Integrals In view of the definition of the definite integral we may approximate its value by choosing the decomposition D to be a decomposition of the interval [a,b] into subintervals of length (b-a)/n for some positive integer n. The points  j can be freely chosen according to any rule from the intervals [x j-1,x j ]. In left rule approximations,  j =x j-1. In mid rule approximations,  j =(x j-1 + x j )/2. In right rule approximations,  j =x j.

5. Mika Seppälä: Numerical Integration Formulae for Approximations

6. Mika Seppälä: Numerical Integration Trapezoidal approximations and Simpson’s Formula Depending on the shape of the function in question, the following approximations are usually better: Trapezoidal Approximation: TRAP(n) = (LEFT(n)+RIGHT(n))/2 Simpson’s Approximation: SIMPSON(n)=(2MID(n)+TRAP(n))/3.

7. Mika Seppälä: Numerical Integration Properties of Approximations If f is strictly increasing – like in the above picture – then the above inequalities are also strict. If f is decreasing, then the direction of the above inequalities must be changed.

8. Mika Seppälä: Numerical Integration Properties of Approximations These estimates show that the approximations can be made as precise as needed simply by increasing the number n of subintervals.

9. Mika Seppälä: Numerical Integration Properties of Midpoint Approximations A function which is concave up has the property that its graph lies above any tangent line. This observation leads to the following estimate valid for functions that are concave up. The blue triangle on the right has been obtained by letting the top side of the rectangle on the left turn around the point where it intersects the graph of the function f. Since this is also the midpoint of the top side, the areas of the two blue domains are the same.