$Step=%v$

$定義:\;f(x)≥0且F(x)=\int_{a}^{x}{f(t)dt }$

$a$

$b$

$x$

$h=%v$

$x+h$

$\int_{a}^{x+h}{ f(t)dt}-\int_{a}^{x}{f(t)dt }$
$=F(x+h)-F(x)$

$M$

$m$

⇒mh≤F(x+h)-F(x)≤Mh

⇒m≤\frac{F(x+h)-F(x)}{h}≤M

⇒\lim_{ h \to 0}m≤f(x)≤\lim_{ h \to 0 }M

F'(x)=\lim_{h \to 0}\frac{F(x+h)-F(x)}{h}=f(x)

⇒F(x)=\int{ f(x)dx}

設G(x)是f(x)的一個反導函數

設F(x)=\int_{a}^{x}{f(t)dt }

F(b)-F(a)=\int_{a}^{b}{ f(t)dt}-\int_{a}^{a}{f(t)dt }

⇒\int_{a}^{b}{ f(t)dt}-0=\int_{a}^{b}{f(x)dx }

又G(x)=F(x)+c

$G(x)|^b_a=G(b)-G(a)$
$=\left(  F(b)+c \right)-\left( F(a)+c  \right)  $
$=F(b)-F(a)=\int_{a}^{b}{ f(x)dx}$

s = 1 (slider from 1 to 16, step 1)
E = (-3.54, 2.34)
F = (0.8, 3.88)
G = (3.42, 2.84)
H = (5.9, 3.7)
f = Polynomial({E, F, G, H})
A = Point(xAxis)
B = Point(xAxis)
C = Point(xAxis)
h = 1.7 (slider from 0.1 to 2, step 0.1)
D: C + (h, 0)
K_1: (x(C), f(x(C)))
K_2: (x(D), f(x(D)))
a = Integral(f, x(C), x(D))
M = Max(f, x(C), x(D))
m = Min(f, x(C), x(D))
k_! = Segment(A, (x(A), f(x(A))))
k_2 = Segment(B, (x(B), f(x(B))))
u_s = UpperSum(f, x(C), x(D), 1)
l_s = LowerSum(f, x(C), x(D), 1)
a_2 = Integral(f, x(C), x(A))
Title: "微積分第" + (LaTeX[If[s ≤ 10, "一", "二"]]) + "基本定理"