積分の応用
1 不定積分とその基本公式
\[例題1 次の関数を積分せよ。\\2 定積分
\[例題1 次の定積分をせよ。\\3 面積・体積
\[例題1 -\pi\le x\le\piにおいて,2曲線 y=\sin x,y=\cos xで囲まれた図形の面積を求めよ。\]積分の応用
1 不定積分とその基本公式
\[例題1 次の関数を積分せよ。\\--> | integrate((x^(1/3)+3)/(x), x); |
--> | integrate((x*sqrt(2*x-1)),x); |
--> | factor(%); |
--> | integrate(2*x*sqrt(x^2+1),x); |
--> | integrate(log(x),x); |
--> | integrate((x^2+3)/(x+1),x); |
--> | integrate((x-3)/(x-1)/(x-2),x); |
--> | integrate((sin(x))^2*cos(x),x); |
--> | integrate((sin(x))^2,x); |
--> | ratsimp(%); |
--> | integrate(sin(4*x)*cos(2*x),x); |
2 定積分
\[例題1 次の定積分をせよ。\\--> | integrate(abs(%e^x-%e), x, -1, 1); |
--> | integrate(sqrt(a^2-x^2),x,0,a); |
--> | integrate(1/(1+x^2),x,0,1); |
--> | integrate(x*sin(x),x,0,%pi/2); |
--> | integrate((x-1)*log(x),x,1,2); |
--> | diff(integrate((x-t)*cos(3*t),t,a,x),x,2); |
--> | f(x):=sin(x)+3*k; |
--> | solve(k=integrate(f(t)*cos(t),t,0,%pi/2)); |
--> | limit(nusum(1/n/sqrt(n)*sqrt(k),k,1,n),n,inf); |
--> | integrate(sqrt(x),x,0,1); |
--> | is(nusum(1/k,k,1,n)>log(n+1)); |
--> | integrate(1/x,x,1,n+1); |
3 面積・体積
\[例題1 -\pi\le x\le\piにおいて,--> | wxplot2d([sin(x),cos(x)], [x,-%pi,%pi])$ |
--> | f1:sin(x); |
--> | f2:cos(x); |
--> | solve(sin(x-pi/4)=0,x); |
--> | [x1,x2]:[-3/4*%pi,%pi/4]; |
--> | load(draw)$ |
--> |
draw2d( fill_color=cyan, filled_func=f2, explicit(f1,x,x1,x2), filled_func=false, xaxis=true, yaxis=true, explicit(f1,x,-%pi,%pi), explicit(f2,x,-%pi,%pi) )$ |
--> | integrate(f2-f1,x,-3*%pi/4,%pi/4); |
--> | kill(f1,f2,x1,x2)$ |
--> |
barea(f1,f2):=block( [x1,x2]:map('rhs,solve(f1-f2)), if x1 > x2 then (c:x1,x1:x2,x2:c), draw2d( xaxis=true, yaxis=true, fill_color=cyan, filled_func=f1, explicit(f2,x,x1,x2), filled_func=false, explicit(f1,x,x1,x2), explicit(f2,x,x1,x2) ), return(abs(integrate(f1-f2,x,x1,x2))) )$ |
--> | barea(4*x-x^2,x^2/3); |
--> | integrate(2*b/a*sqrt(a^2-x^2),x,-a,a); |
--> | 4*integrate(a*b*sin(x)^2,x,0,%pi/2); |
--> |
barea1(f1,f2,t1,t2):=block( x1:subst(t1,t,f1),x2:subst(t2,t,f1), if x1>x2 then (c:x1,x1:x2,x2:c), wxplot2d([['parametric, f1,f2, [t, t1, t2], [nticks, 300]]], [x,x1,x2], [plot_format, gnuplot]), return(abs(integrate(f2*diff(f1,t),t,t1,t2))) )$ |
--> | a:1$ |
--> | wxplot2d([['parametric, a*(t-sin(t)), a*(1-cos(t)), [t, -%pi/2, 2.5*%pi], [nticks, 300]]], [x,-1,7])$ |
--> | kill(a)$ |
--> |
barea2(f1,f2,t1,t2):=block( x1:subst(t1,t,f1),x2:subst(t2,t,f1), if x1>x2 then (c:x1,x1:x2,x2:c), return(abs(integrate(f2*diff(f1,t),t,t1,t2))) )$ |
--> | barea2(a*(t-sin(t)),a*(1-cos(t)),0,2*%pi); |
--> |
draw3d( implicit(x^2+y^2=9,x,-1,3,y,-3,3,z,0,6), explicit(-y,x,-1,3,y,-3,0), surface_hide = true, color=blue, implicit(x=-1,x,-3,3,y,-3,0,z,0,3) )$ |
--> | integrate((sqrt(3^2-x^2))^2/2,x,-3,3); |
--> |
xrot(f,a,b):=block( return(%pi*integrate(f^2,x,a,b)) )$ |
--> | 2*xrot(sqrt(r^2-x^2),0,r); |
--> | b:2$r:1$ |
--> |
draw3d( color=blue, surface_hide = true, parametric_surface(r*sin(s),(r*cos(s)+b)*sin(t),(r*cos(s)+b)*cos(t), s, 0, 2*%pi, t, 0, 2*%pi) )$ |
--> | kill(b,r)$ |
--> | 2*%pi*integrate((sqrt(r^2-x^2)+b)^2-(-sqrt(r^2-x^2)+b)^2,x,0,r); |
--> |
zRevolution(f):=block( draw3d( title = "z=f(x) Revolution", color=blue, surface_hide = true, parametric_surface(x*cos(s),x*sin(s), f,s,0,2*%pi,x,0,1) ) )$ |
--> | zRevolution(x^2/2); |
--> |
yrot(g,a,b):=block( return(%pi*integrate(g^2,y,a,b)) )$ |
--> | yrot(sqrt(2*y),0,a); |
--> | integrate(2*%pi*x*x^2/2,x,0,sqrt(2*a)); |
--> | a:k^2; |
--> | integrate(2*%pi*x*x^2/2,x,0,sqrt(2*a)); |
--> | kill(a)$ |
--> | wxplot2d(1-sqrt(x),[x,0,1])$ |
--> | zRevolution(1-sqrt(x)); |
--> |
xRevolution(f):=block( draw3d( title = "x=f(y) Revolution", color=blue, surface_hide = true, parametric_surface(f,y*sin(s),y*cos(s),s,0,2*%pi,y,0,1) ) )$ |
--> | xRevolution((1-y)^2); |
--> |
xyrot(f,a,b):=block( return(2*%pi*integrate(f*x,x,a,b)) )$ |
--> | xrot(1-sqrt(x),0,1); |
--> | yrot((1-y)^2,0,1); |
--> | xyrot(1-sqrt(x),0,1); |
積分の応用
1 不定積分とその基本公式
原始関数:primitive function2 定積分
定積分:difinit integral3 面積・体積
円環体:torus