Please begin with general idea of graphing and the codes

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% GAUSSIAN DOUBLE INTEGRAL ALGORITHM 4.5 % % To approximate I = double integral (( f(x, y) dy dx )) with limits % of integration from a to b for x and from c(x) to d(x) for y: % % INPUT: endpoints a, b; positive integers m, n. (Assume that the % roots r(i,j) and coefficients c(i,j) are available for % i equals m and n for 1 B fprintf(1,'Lower limit must be less than upper limit '); else OK = TRUE; end end OK = FALSE; while OK == FALSE fprintf(1,'Input two integers M > 1 and N > 1 on separate lines. '); fprintf(1,'This implementation of Gaussian quadrature requires '); fprintf(1,'both to be less than or equal to 5. '); fprintf(1,'M is used for the outer integral and N for the inner '); fprintf(1,'integral. '); M = input(' '); N = input(' '); if M 5 | N > 5 fprintf(1,'Integers must be less than or equal to 5. '); else OK = TRUE; end end end r = zeros(4,5); co = zeros(4,5); if OK == TRUE r(1,1) = 0.5773502692; r(1,2) = -r(1,1); co(1,1) = 1.0; co(1,2) = 1.0; r(2,1) = 0.7745966692; r(2,2) = 0.0; r(2,3) = -r(2,1); co(2,1) = 0.5555555556; co(2,2) = 0.8888888889; co(2,3) = co(2,1); r(3,1) = 0.8611363116; r(3,2) = 0.3399810436; r(3,3) = -r(3,2); r(3,4) = -r(3,1); co(3,1) = 0.3478548451; co(3,2) = 0.6521451549; co(3,3) = co(3,2); co(3,4) = co(3,1); r(4,1) = 0.9061798459; r(4,2) = 0.5384693101; r(4,3) = 0.0; r(4,4) = -r(4,2); r(4,5) = -r(4,1); co(4,1) = 0.2369268850; co(4,2) = 0.4786286705; co(4,3) = 0.5688888889; co(4,4) = co(4,2); co(4,5) = co(4,1); % STEP 1 H1 = (B-A)/2; H2 = (B+A)/2; % use AJ in place of J AJ = 0; % STEP 2 for I = 1:M % STEP 3 X = H1*r(M-1,I)+H2; JX = 0; C1 = C(X); D1 = D(X); K1 = (D1-C1)/2; K2 = (D1+C1)/2; % STEP 4 for J = 1:N Y = K1 * r(N-1,J)+K2; Q = F(X, Y); JX = JX + co(N-1,J)*Q; end % STEP 5 AJ = AJ+co(M-1,I)*K1*JX; end % STEP 6 AJ = AJ*H1; % STEP 7 fprintf(1,' The double integral of F from %12.8f to %12.8f is ', A, B); fprintf(1,' %.10e', AJ); fprintf(1,' obtained with M = %3d and N = %3d ', M, N); end

THE ABOVE IS THE READY TO USE CODE.

5. (10 points) Solve the following problems using computer. Use Algorithm from the Lec- ture Notes "Multiple Integrals" or Algorithm 4.4 in the textbook with 1) n = 4, m = -8, 2) n = 8, m = 4, and 3) n = m = 6 to approximate the following double integral and compare the results to the exact solution . CL cos ydydx. (1) 5. (10 points) Solve the following problems using computer. Use Algorithm from the Lec- ture Notes "Multiple Integrals" or Algorithm 4.4 in the textbook with 1) n = 4, m = -8, 2) n = 8, m = 4, and 3) n = m = 6 to approximate the following double integral and compare the results to the exact solution . CL cos ydydx. (1)