In this problem, you will develop another way to think about the trapezoidal rule

Chapter 5, Problem 26

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In this problem, you will develop another way to think about the trapezoidal rule approximation given in equation (6). (a) Let L denote a general linear function of two variables, that is, L(x, y) = Ax + By + C, where A, B, and C are constants. Set R = [a, b] [c, d]. Show that R LdA = (area of R)(average of the values of L taken at the four vertices of R). (Note that this gives an exact expression for the double integral.) (b) Suppose that f is any function of two variables that is integrable on R. Show that the trapezoidal rule approximation T1,1 to R f dA is T1,1 = (area of R)(average of the values of f taken at the four vertices of R). (c) Now let x = (b a)/m, y = (d c)/n, and, for i = 1,..., m, j = 1,..., n, let Ri j = [xi1, xi] [yj1, yj], where xi = a + ix and yj = c + jy. Then we have R f dA = n j=1 m i=1 Ri j f dA. Use T1,1 to approximate each integral Ri j f dA and sum the results to obtain the formula for Tm,n given by equation (6). True/False Ex