第五章 定积分

说明：为了复习高数，该文章是学习课程 《高等数学》同济版 全程教学视频（宋浩老师）而记录的笔记，笔记来源于本人。若有侵权，请联系本人删除。笔记难免可能出现错误或笔误，若读者发现笔记有错误，欢迎在评论里批评指正。参考书籍： 高等数学上册(同济_第7版)。

一、定积分的概念（定义、几何意义、近似计算）、性质

二、微积分的基本公式（积分上限函数、牛顿-莱布尼茨公式）

三、定积分的换元法

画例（5）两个图的代码需要两个.m文件，一个文件用于放画阴影的函数dohatch，该函数的原代码的链接为：MATLAB怎么实现对一个函数的特定区间加上阴影。另一个文件用于画图，是主文件。
函数dohatch的代码:

function h = dohatch(x,y,angle,color,style,step,width)
%DOHATCH  Hatches a two-dimensional domain.
%   DOHATCH(x,y,ANG,COL,'Style',D,W) is similar to the FILL command
%   but fills the closed contour with hatch lines instead of uniform
%   color. Vectors x and y are the coordinates of the boundary line of
%   the domain to be hatched. Scalar ANG is the slope of the hatches
%   (in degrees). COL is the color of the hatching which could both be
%   a 1-by-3 vector ([red green blue] ), or a color specifier ('r','g',
%   'b', 'w','y','c','m'). 'Style' specifys the linestyle ('-','--','-.',
%   ':'). And also, D is the steps (distances between hatches), W the
%   linewidth (thickness) of the hatch lines (the last two in points).
%
%   H = DOHATCH(...) returns the handle of the hatching.
%
%   Note:
%   This is a simplified version of the function HATCH (File ID: #2075)
%   in MathWorks FileExchange website, running in manual style instead of
%   in automatic style, because the original version still has a few bugs
%   under some circumstances.
%
%   See also: FILL, LINE.

%   Revised by Kastin
%   June, 28, 2015

if nargin>7||nargin<2
    error('MATLAB:dohatch:WrongNumOfInputs',...
        'Wrong number of inputs.')
end

% Here we do not check the validation of inputs any more for clearity. 

% Defaults
angledflt = 45;        % Angle in degrees
colordflt = [1 1 1];   % Color
styledflt = '-';       % Linestyle
widthdflt = 1;         % Thickness of the lines
stepdflt = 10;         % Distance between hatches

if nargin<7, width = widthdflt; end
if nargin<6, step = stepdflt;   end
if nargin<5, style = styledflt; end
if nargin<4, color = colordflt; end
if nargin<3, angle = angledflt; end

angle = angle*pi/180;             % Degrees to radians
x=x(:).'; y=y(:).';
yi = find(~isnan(x)&~isnan(y));
x = x(yi); y = y(yi);             % Remove NaN's
x = [x x(1)]; y = [y y(1)];       % Close loop
ll = length(x);

% Transform the coordinates
oldu = get(gca,'units');
set(gca,'units','points')
sza = get(gca,'pos'); sza = sza(3:4);
xlim = get(gca,'xlim');
ylim = get(gca,'ylim');

islx = strcmp(get(gca,'xscale'),'log');
isly = strcmp(get(gca,'yscale'),'log');
if islx     % If log scale in x
	xlim = log10(xlim);
	x = log10(x);
end
if isly     % If log scale in y
	ylim = log10(ylim);
	y = log10(y);
end

xsc = sza(1)/(xlim(2)-xlim(1)+eps);
ysc = sza(2)/(ylim(2)-ylim(1)+eps);

ca = cos(angle); sa = sin(angle);
x0 = mean(x); y0 = mean(y);  % Central point
x = (x-x0)*xsc; y = (y-y0)*ysc;
yi = x*ca+y*sa;              % Rotation
y = -x*sa+y*ca;
x = yi;
y = y/step;    % Make steps equal to one

% Compute the coordinates of the hatch line
yi = ceil(y);
ll = length(y);
yd = [yi(2:ll)-yi(1:ll-1) 0];
dm = max(abs(yd));
fnd = find(yd);
lfnd = length(fnd);
A = sign(yd(fnd));
edm = ones(dm,1);
A = A(edm,:);
if size(A,1)>1, A = cumsum(A); end
fnd1 = find(abs(A)<=abs(yd(edm,fnd)));
A = A+yi(edm,fnd)-(A>0);
xy = (x(fnd+1)-x(fnd))./(y(fnd+1)-y(fnd));
xi = x(edm,fnd)+(A-y(edm,fnd)).*xy(edm,:);
yi = A(fnd1);
xi = xi(fnd1);

% Sorting points of the hatch line ........................
li = length(xi); 
xi0 = min(xi); xi1 = max(xi);
yi0 = min(yi); yi1 = max(yi);
ci = yi*(xi1-xi0)+xi;
[ci,num] = sort(ci);
xi = xi(num); yi = yi(num);
if floor(li/2)~=li/2
        xi = [xi xi(li)];
        yi = [yi yi(li)];
end

% Organize to pairs and separate by  NaN's
li = length(xi);
xi = reshape(xi,2,li/2);
yi = reshape(yi,2,li/2);
xi = [xi; ones(1,li/2)*nan];
yi = [yi; ones(1,li/2)*nan];
xi = xi(:)'; yi = yi(:)';

% Transform to the original coordinates
yi = yi*step;
xy = xi*ca-yi*sa;
yi = xi*sa+yi*ca;
xi = xy/xsc+x0;
yi = yi/ysc+y0;

hl = line('xdata',xi,'ydata',yi,...
    'color',color,'linestyle',style,'linewidth',width);
set(gca,'units',oldu)   % Set axes units back

if nargout>0, h=hl; end
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主文件的代码：

%MATLAB代码
%第一张图
x=-4:0.001:4;y=1/2*x.^2;
figure(1)
hold on;
plot(x,y);plot([-3.5 -3.5],[0 1/2*(-3.5).^2],'--');plot([3.5 3.5],[0 1/2*3.5.^2],'--');%画图
%将坐标轴放到原点
ax = gca;
ax.XAxisLocation = 'origin'; 
ax.YAxisLocation = 'origin'; 
set(gca,'xtick',-5:1:5,'ytick',-1:1:9);%刻度间隔设为1
axis equal;%同样的刻度范围显示出来是等长的
xlabel('x');ylabel('y');%给出x、y轴
legend("y_1=1/2*x^2","x_2=-3.5","x_3=3.5");
%在函数的一个区间内填充阴影
idx=x>=-3.5&x<=3.5;
xi=x(idx);
dohatch([xi xi(end) xi(1)],[y(idx) 0 0],30,'b','-.',8,1)
hold off

%第二张图
x1=-4:0.001:4;y1=1/7*x1.^3;
figure(2)
hold on;
plot(x1,y1);plot([-3.5 -3.5],[0 1/7*(-3.5).^3],'--');plot([3.5 3.5],[0 1/7*3.5.^3],'--');%画图
%将坐标轴放到原点
ax = gca;
ax.XAxisLocation = 'origin'; 
ax.YAxisLocation = 'origin'; 
set(gca,'xtick',-10:1:10,'ytick',-10:1:10);%刻度间隔设为1
axis equal;%同样的刻度范围显示出来是等长的
xlabel('x');ylabel('y');%给出x、y轴
legend("y_1=1/7*x^3","x_2=-3.5","x_3=3.5");
%在函数的一个区间内填充阴影
idx1=x1>=0&x1<=3.5;
x1i=x1(idx1);
dohatch([x1i x1i(end) x1i(1)],[y1(idx1) 0 0],30,'b','-.',8,1)
idx1=x1>=-3.5&x1<=0;
x1i=x1(idx1);
dohatch([x1i x1i(end) x1i(1)],[y1(idx1) 0 0],-30,'b','-.',8,1)
hold off
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四、分部积分

五、反常积分（无穷限、无界函数的反常积分）、伽马函数