A Fredholm integral equation ofthe second kind is an equation ofthe form u(x) = f(x)+ j K(x,t)u{t) dt, Ja where a and b and the functions / and K are given. To approximate the function u on the interval [a, h], a partition xq = a < xj < < xm_i < x, = h is selected, and the equations u(x,) = /(x,) + K(x,-, t)u(t) dt, for each / = 0,..., m, In are solved for m(xo), m(xi), ..., u(xm). The integrals are approximated using quadrature formulas based on the nodes Xo,..., xm. In our problem, a = 0, b = I, f(x) = x 2 , and K(x, t) = e 1* - ' 1 . a. Show that the linear system M(0) = /(0) + i[K(0. 0)H(0) + K(0, 1M1)1, (1) = /(I) + ^[/f(l, 0)(0) + (1, l)H(l)] must be solved when the Trapezoidal rule is used. b. Set up and solve the linearsystem that results when the Composite Trapezoidal rule is used with 77 - 4. c. Repeat part (b) using the Composite Simpson rule.

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