Numerical integration is the process of approximating a definite integral using well-chosen sums of function values. It is needed when we cannot find an antiderivative explicitly, as in the case of the Gaussian function To approximate the definite integral we fix a whole number N and divide [a, b] into N subintervals of length Δx = (b −a)/N. The endpoints of the subintervals arewhole number We shall denote the values of f (x) at these endpoints by y j : In particular, y 0 = f (a) and y N = f (b).

Trapezoidal Rule The Nth trapezoidal approximation to CONCEPTUAL INSIGHT We see that the area of the jth trapezoid is equal to the average of the areas of the endpoint rectangles with heights y j −1 and y j. It follows that T N is equal to the average of the right- and left-endpoint approximations R N and L N introduced in Section 5.1:Section 5.1 In general, this average is a better approximation than either R N alone or L N alone.

Use a trapezoidal sum to approximate where f has the given values.

Midpoint Rule The Nth midpoint approximation to GRAPHICAL INSIGHT M N has a second interpretation as the sum of the areas of tangential trapezoids—that is, trapezoids whose top edges are tangent to the graph of f (x) at the midpoints c j. The trapezoids have the same area as the rectangles because the top edge of the trapezoid passes through the midpoint of the top edge of the rectangle. The rectangle and the trapezoid have the same area.

Calculate T 6 and M 6 for