目录

一、复化梯形公式求积分1.1 题目1.2 程序1.3 运行结果 二、复化梯形公式求二重积分2.1 题目2.2 程序2.2.1 二重积分通用程序2.2.2 主函数程序 2.3 运行结果

一、复化梯形公式求积分

1.1 题目

利用复化梯形公式计算

I ( f ) = ∫ 1 5 s i n x x d x I(f) = \int_1^5{\frac{sinx}{x}dx} I(f)=∫15​xsinx​dx

的近似值，精确至7位有效数字。

1.2 程序

clc; clear allsyms x %% 已知条件Fx(x) = sin(x)/x;a = 1;b = 5;epsilon = 5e-8;e = 1;T = []; % 存放Tf的值h = b-a;Xk1 = [a, b]; % 存放已选的 xkXk2 = [];% 存放将要使用的 x(k+1/2)sumfx = 0;% sum( fx(k+1/2) )%% 计算digits(8)format longk = 0;Tf = vpa( b - a ) / 2 * ( Fx(a) + Fx(b) );T = [T, Tf];fprintf('k=%d, 2^k=%d, Tk=%0.8f\n',k,2^k,Tf) while(e>epsilon)for i=1:length(Xk1)-1temp = ( Xk1(i) + Xk1(i+1) ) / 2;sumfx = sumfx + Fx(temp);Xk2 = [Xk2,temp];endXk1 = [Xk1,Xk2];Xk1 = sort(Xk1);k = k + 1;Tf = vpa( 0.5 * ( Tf + h * sumfx ) );T = [T, Tf];e = abs(Tf - T(end-1))/3;fprintf('k=%d, 2^k=%d, Tk=%0.8f, e=%0.8f\n',k,2^k,Tf,e) % 更新值h = h / 2;sumfx = 0;Xk2 = [];endIf = Tf;fprintf('复化梯形求积公式的结果为：%0.8f\n',If)

1.3 运行结果

二、复化梯形公式求二重积分

2.1 题目

（1）给定积分 I ( f ) = ∫ c d ( ∫ a b f ( x , y ) d x ) d y I(f)=\int_{c}^{d}\left(\int_{a}^{b} f(x, y) \mathrm{d} x\right) \mathrm{d} y I(f)=∫cd​(∫ab​f(x,y)dx)dy，取初始步长 h h h和 k k k及精度 ε \varepsilon ε，应用复化梯形公式，采用逐次二分步长的方法并应用外推思想编制计算 I ( f ) I(f) I(f)的通用程序，计算至相邻两次近视值之差的绝对值不超过 ε \varepsilon ε为止；

（2）用所编程序计算积分 I ( f ) = ∫ 0 π / 6 ( ∫ 0 π / 3 tan ⁡ ( x 2 + y 2 ) d x ) d y I(f)=\int_{0}^{\pi / 6}\left(\int_{0}^{\pi / 3} \tan \left(x^{2}+y^{2}\right) \mathrm{d} x\right) \mathrm{d} y I(f)=∫0π/6​(∫0π/3​tan(x2+y2)dx)dy

取 ε = 1 2 × 1 0 − 5 \varepsilon=\frac{1}{2} \times 10^{-5} ε=21​×10−5。

2.2 程序

2.2.1 二重积分通用程序

function[result] = trapezoid(a,b,c,d,F,epsilon)%% 初值条件T0 = []; % 存放Tm,n(f)T1 = []; % 存放T(1)m,n(f)T2 = []; % 存放T(2)m,n(f)T3 = []; % 存放T(3)m,n(f)m = 1; % 将[a,b] m等分n = 1; % 将[c,d] n等分; 注：虽然这里m与n相等，但分开计算。h = (b - a) / m; % x的步长k = (d - c) / n; % y的步长xx = [a,b];yy = [c,d];e = 1; % 误差%% 计数位count = 0;T0_count = 0;T1_count = 0;T2_count = 0;T3_count = 0;%% 计算digits(10)T = h * k / 4 * ( F(xx(1),yy(1)) + F(xx(1),yy(2)) + F(xx(2),yy(1)) + F(xx(2),yy(2)) ) ;T0 = [T0, T];T0_count = T0_count + 1;count = 1;fprintf('k=%d, 2^k=%d, T0=%0.8f\n',count,2^count,T) while(e>epsilon)m = 2 * m;n = 2 * n;h = (b - a) / m;k = (d - c) / n;% 计算x、y的取值xx = [a]; % 暂时只存入左边值yy = [c]; % 暂时只存入左边值for i = 1:mtemp = a + i * h;xx = [xx,temp];endfor j = 1:ntemp = c + j * k;yy = [yy,temp];end% 计算T0T = 0;for i = 1:mfor j = 1:nT = T + vpa( h * k /4 * (F(xx(i),yy(j)) + F(xx(i),yy(j+1)) + F(xx(i+1),yy(j)) + F(xx(i+1),yy(j+1)) ) );endendT0 = [T0,T];T0_count = T0_count + 1;count = count + 1;fprintf('k=%d, 2^k=%d, T0=%0.8f',count,2^count,T) % 计算T1if T0_count - T1_count == 2 && T0_count >= 2temp = 4 / 3 * T0(end) - 1 / 3 * T0(end-1);T1 = [T1,temp];T1_count = T1_count + 1;fprintf(', T1=%0.8f',T1(end)) end% 计算T2if T1_count - T2_count == 2 && T1_count >= 2temp = 4 / 3 * T1(end) - 1 / 3 * T1(end-1);T2 = [T2,temp];T2_count = T2_count + 1;fprintf(', T2=%0.8f',T2(end)) end% 计算T3if T2_count - T3_count == 2 && T2_count >= 2temp = 4 / 3 * T2(end) - 1 / 3 * T2(end-1);T3 = [T3,temp];T3_count = T3_count + 1;fprintf(', T3=%0.8f',T3(end)) endif T3_count >= 2e = T3(end) - T3(end-1);fprintf(', e=%0.8f',e) endfprintf('\n') endfprintf('复化梯形二重积分求积公式的结果为：%0.8f\n',T3(end))result = T3(end);end

2.2.2 主函数程序

clc; clear;syms x yF(x,y) = tan(x^2 + y^2);a = 0;b = pi/3;c = 0;d = pi/6;epsilon = 0.5e-5;result = trapezoid(a,b,c,d,F,epsilon);

2.3 运行结果