SKEDSOFT

Evaluation of Double Intergral Using Trapezium Rule : The Trapezium Method is also used in Evaluating a Double Integral.Evaluating a multiple integral involves expressing it as an iterated integral, which can then be evaluated either symbolically or numerically. We begin by discussing the evaluation of iterated integrals.

We consider the evaluation of the double integral

.....................1.1

over a rectangle x = a, x = b, y = c, y = d.

Figure 1.1

Evaluating the inner integral in (1.1) by trapezium rule, we obtain

........................1.2

Using the trapezium rule again to evaluate the integrals in (1.2), we obtain

.......................1.3

Notice that the points (a, c), (a, d), (b, c), (b, d) are the four corners of the rectangle (Fig. 1.1).

If we denote h = b – a, k = d – c, we can write the formula as

...................1.4

The weights (coefficients of the ordinates f ) in the trapezium rule are given in the computational molecule figure 1.2

Figure:1.2

  Composite trapezium rule :

Divide the interval [a, b] into N equal subintervals each of length h, and the interval [c, d] into M equal subintervals each of length k. We have

The general grid point is given by (x_i, y_j). Denote, f_ij = f(x_i, y_j). That is,