微积分第二定理

Info about F’

$\Delta F=F(b)-F(a),\Delta x=b-a$

$\Delta F={\int }_a^{b}f(x)dx(FTC1)$

$\frac{\Delta F}{\Delta x}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$

Average (f)

FTC2

IF f is continuous,and $G(x)={\int }_a^{x}f(t)dt;a\le t\le x$

then G’(x) = f(x)

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定积分在对数和集合中的应用

FTC2:

$\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$

$Solvey=\frac{1}{x}$

Defination of Log:

$L(x)={\int }_1^x\frac{dt}{t}$

Claim： L(ab) = L(a) + L(b)

Fresnel:

$C(x)={\int }_0^{x}cos(t^2)dt$

$S(x)={\int }_0^{x}sin(t^2)dt$

几何绘图法 AREAS BETWEEN CURVES

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壳层法，圆盘法面积

SLICE切片：

$\Delta V\approx A\Delta x$

dv = A(x) dx

$V=\int A(x)dx\approx \sum A_i\Delta x$

Solids of revolution 旋转立方体

壳层法|Disks|:

$dV=(\pi y^2)dx$

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功，平均值，概率

Average value

$\frac{y_1+...+y_n}{n}\to \frac{1}{b-a}{\int }_a^{b}f(x)dx$

Continous average = AVE(f)

y=f(x)

$\Delta x=\frac{b-a}{n}$

spacing

a= x_0<x_1<x_2<…<x_n=b

y_1=f(x_1),y_2=f(x_2)…y_n=f(x_n)

Riem Sum

$\frac{(y_1+...+y_n)\Delta x}{b-a}\underset{\Delta x\to 0}{\to }\frac{{\int }_a^{b}f(x)dx}{b-a}$

$\frac{\Delta x}{b-a}=\frac{1}{n}\to 0(n\to \infty )$

WEIGHTED AVERAGE

$\frac{{\int }_a^{b}f(x)w(x)dx}{{\int }_a^{b}w(x)dx}=f(x)$