MATLAB_积分法

MATLAB积分法•数值积分法1.矩形法:sum();x=0:pi/1000:pi;y=sin(x.^2);v=sum(y(1:end-1))*pi/1000v=0.773320sinVxdx2.梯形法:trapz():x=0:pi/1000:pi;y=sin(x.^2);v=trapz(x,y)v=0.7726或者:v=trapz(y)*pi/1000v=0.77263.辛卜生法:quad(fun,a,b,tol);求函数fun在区间[a,b]上的积分,tol为误差精度.fun=inline('sin(x.^2)','x');quad(fun,0,pi)ans=0.7727默认误差精度10-6,也可自由设定.quad(fun,0,pi,1e-10)ans=0.7727dblquad(fun,xmin,xmax,ymin,ymax,tol),求二重积分.如:fun=inline('x.^y','x','y');dblquad(fun,0,1,1,2)ans=0.4055triplequad(fun,xmin,xmax,ymin,ymax,zmin,zmax,tol)求三重积分.2110yVxdxdy求积分:f=inline('(x.^2+y.^2).*(x=y)','x','y');dblquad(f,0,1,0,1)ans=0.333312200yVxydxdy•符号积分法1.符号的生成及运算:syms;生成符号symsarg1arg2…argv其中:argv=‘positive’表示正实数,‘real’表示实数,‘unreal’表示非实数.symsa11a12a21a22A=[a11a12;a21a22]A=[a11,a12][a21,a22]det(A)ans=a11*a22-a12*a21r=collect(f,x)合并同类项.symsxf=(x-1)*(x-2);r=collect(f,x)r=2+x^2-3*xf=factor(r)因式分解.z=factor(r)z=(x-1)*(x-2)r=expand(s)把表达式展开.f=cos(2*x)*sin(x);expand(f)ans=2*sin(x)*cos(x)^2-sin(x)2.符号积分R=int(S,x,a,b)定积分;R=int(S,x)不定积分.如:symsxf=x^2+x+1;r=int(f,x,0,1)r=11/61201Vxxdxint(f,x)ans=1/3*x^3+1/2*x^2+xsymsxyf=x*y+y;int(f,x)ans=1/2*x^2*y+x*yint(f,y)ans=1/2*x*y^2+1/2*y^2symsxabf=x^2+1;int(f,x,a,b)ans=1/3*b^3-1/3*a^3+b-a多重积分:2110yVxdxdysymsxyf=x^y;int(int(f,x,0,1),y,1,2)ans=log(3)-log(2)symsxyf=x^2+y^2;int(int(f,x,0,y),y,0,1)ans=1/312200yVxydxdy平衡状态中气体分子速率分布函数:f(v)称为麦克斯韦速率分布函数.平均速率为速率v的数学期望:设则:23222()42mvkTmfvevkT0()vvfvdv2mbkT232304bvbvvedvsymsvbpositivef=v^3*exp(-b*v^2);int(f,v,0,inf)ans=1/2/b^2322118422bkTvbbm