1. 定义

$$\begin{gathered} z=f(x,y)\text{vol}={\iint }_{R}f(x,y)\mathrm{d}A \\ \text{vol}={\int }_{x_{min}}^{x_{max}}S(x)\mathrm{d}x \\ \text{For given x, }S(x)={\int }_{y_{min}(x)}^{y_{max}(x)}f(x,y)\mathrm{d}y \\ \Rightarrow \text{vol}={\int }_{x_{min}}^{x_{max}}{\int }_{y_{min}(x)}^{y_{max}(x)}f(x,y)\mathrm{d}y\mathrm{d}x \end{gathered}$$
计算二重积分时，利用切面，将二重积分转化为两个单变量积分

附录

附录1. 二重积分

$$\begin{gathered} z=-x^2-y^2+1 \\ {\int }_0^1{\int }_0^{\sqrt{1-x^2}}(-x^2-y^2+1)\mathrm{d}y\mathrm{d}x=\frac{\pi }{8} \end{gathered}$$

clear
clc
clf

inte = 0.05;
X = 0:inte:1-inte;
j = 1;
for i=X
    disp(i);
    
    subplot(1,2,1);

    f = @(x,y,z) x.^2 + y.^2 + z - 1;
    interval = [-5 5 -5 5 0 5];
    fimplicit3(f,interval,'FaceAlpha',.8);

    hold on
    
    % 画切面
    f = @(x,y,z) x - i;
    interval = [0 1 0 1 0 2];
    fimplicit3(f,interval);
    
    axis([-2 2 -2 2 0 4]);
    axis vis3d
    xlabel('x轴');
    ylabel('y轴');
    zlabel('z轴');
    
    j = j + 1;
%     if (j ~= length(X))
        hold off
%     end
    
    subplot(1,2,2);
    % -1 <= x <= 1, -1 <= y <= 0 的积分
    XA = i:0.01:i+inte;
    YA = 0:0.01:1;
    [XAA, YAA] = meshgrid(XA, YA);
    ZAA = - XAA.^2 - YAA.^2 + 1;
    ZAA(1,1) = 0;
    meshz(XAA,YAA,ZAA);
    
    hold on

    axis([-2 2 -2 2 0 4]);
    axis vis3d
    xlabel('x轴');
    ylabel('y轴');
    zlabel('z轴');
    
    M(j) = getframe;

end

% movie2gif(M, 'iint.gif', 'LoopCount', 0, 'DelayTime', 0);