曲線間的面積(2)
Author: 藍邦偉  老師

兩曲線間的面積

$ \int_{a}^{b}{ f(x)dx} $

$ \int_{a}^{b}{g(x)dx } $

a

b

$ \int_{a}^{b}{[f(x)-g(x)] }dx $

y=f(x)

y=g(x)

f(x)>g(x)>0

f(x)>0>g(x)

0>f(x)>g(x)

$case%v$

1

3

1

2

3

1

$ y=f(x) $

$ y=g(x) $

y=f(x)

$ y=g(x) $

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.75 * Corner[1] + 0.25 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
A_1 = (-1.724645479384277, 3.4412178978945462)
A_2 = (0.41434077603441183, 4.147847285845363)
A_3 = (3.279054510970156, 3.8613759123517886)
A_4 = (5.933689238677279, 4.262435835242792)
f_1 = Polynomial({A_1, A_2, A_3, A_4})
A_5 = (-1.8010378456492302, 2.2380381292215348)
A_6 = (3.431839243500062, 1.6268991991019095)
A_7 = (4.635019012173075, 2.467215228016394)
A_8 = (6.583024351929381, 3.097452249702257)
f_2 = Polynomial({A_5, A_6, A_7, A_8})
L = Point(u)
R = Point(u)
I_1 = Integral(f_1, x(L), x(R))
I_2 = Integral(f_2, x(L), x(R))
n_{1a} = false
n_{1b} = false
n_{1c} = false
I_3 = Integral(f_2, x(L), x(R))
n = 0 (slider from 0 to 3, step 1)
A_0 = (-5.868931349257988, 4.18604346897784)
B_2 = (-1, 3)
B_3 = (1.8657957350685221, 1.3022316424758598)
B_4 = (5.0933732097627935, 2.658196143678778)
B_1 = (8.760206790480545, 4.9308690400611335)
f_3 = Polynomial({B_1, B_2, B_3, B_4})
B_5 = (-1, -3)
B_6 = (1.3119510796476117, -2.0208362900496013)
B_7 = (4.424940004944453, -1.982640106917125)
B_8 = (8.932089614576691, -1.4096973599299765)
f_4 = Polynomial({B_5, B_6, B_7, B_8})
I_4 = Integral(f_3, x(L), x(R))
I_5 = Integral(f_4, x(L), x(R))
I_6 = Integral(f_4, x(L), x(R))
n_{2a} = false
n_{2c} = false
C_1 = (-0.7124466263736475, -1.1996183527013553)
C_2 = (3.3172506941026323, -0.8940488876415428)
C_3 = (5.80000259771361, -1.2760107189663084)
C_4 = (9.676915185659984, -0.7985584298103513)
f_5 = Polynomial({C_1, C_2, C_3, C_4})
C_5 = (-0.5978580769762177, -2.9757408683615156)
C_6 = (1.3883434459125648, -1.390599268363738)
C_7 = (4.787803744702981, -2.0208362900496013)
C_8 = (9.008481980841642, -2.5555828539042733)
f_6 = Polynomial({C_5, C_6, C_7, C_8})
I_7 = Integral(f_5, x(L), x(R))
I_8 = Integral(f_6, x(L), x(R))
n_{3a} = false
n_{3b} = false
n_{3c} = false
I_9 = Integral(f_5, x(L), x(R))
n_1 = false
n_{2b} = false
text1: "R_1=" + (LaTeX[I_1]) + ""
text2: "R_2=" + (LaTeX[I_2]) + ""
text3: "R=R_1-R_2=" + (LaTeX[I_1 - I_2]) + ""
text4: "R_1=" + (LaTeX[I_4]) + ""
text5: "R_2=" + (LaTeX[I_5]) + ""
text6: "$R=R_1-R_2$
     $=R_1+|R_2|=" + (LaTeX[I_4 - I_5]) + "$"
text9: "$ R_1=" + (LaTeX[I_7]) + " $"
text10: "$ R_2=" + (LaTeX[I_8]) + " $"
I_0 = Integral(f_5, x(L), x(R))
text11: "$ R=R_1-R_2 $
$=" + (LaTeX[I_7]) + "-(" + (LaTeX[I_8]) + ")$
$=" + (LaTeX[I_7 - I_8]) + "$"