人工智能教程 - 数学基础课程1.1 - 数学分析（一）18-21 微积分第二定理,应用,壳层法,加权平均
微积分第二定理
Info about F’
ΔF=F(b)−F(a),Δx=b−a\Delta F = F(b) -F(a),\Delta x = b-aΔF=F(b)−F(a),Δx=b−a
ΔF=∫abf(x)dx (FTC1)\Delta F = \int_{a}^{b}f(x) dx \ (FTC1)ΔF=∫abf(x)dx (FTC1)
ΔFΔx=1b−a∫abf(x)dx\color{Red}\frac{\Delta F }{\Delta x}= \frac{1}{b-a} \int_{a}^{b}f(x) dxΔxΔF=b−a1∫abf(x)dx
作者：KuFun人工智能
Average (f)
FTC2 IF f is continuous,and G(x)=∫axf(t)dt; a≤t≤xG(x) = \int_{a}^{x}f(t)dt; \ \ a\leq t\leq xG(x)=∫axf(t)dt; a≤t≤x then G’(x) = f(x) 定积分在对数和集合中的应用FTC2:
ddx∫axf(t)dt=f(x)\frac{d}{dx}\int_{a}^{x}f(t)dt = f(x)dxd∫axf(t)dt=f(x) Solve y=1xSolve \ \ y=\frac{1}{x}Solve y=x1 Defination of Log: L(x)=∫1xdtt{\color{Red} L(x) = \int _{1}^{x} \frac{dt}{t}}L(x)=∫1xtdt Claim： L(ab) = L(a) + L(b) Fresnel:C(x)=∫0xcos(t2)dtC(x) = \int_{0}^{x}cos(t^2)dtC(x)=∫0xcos(t2)dt
S(x)=∫0xsin(t2)dtS(x) = \int_{0}^{x} sin(t^2) dtS(x)=∫0xsin(t2)dt
几何绘图法 AREAS BETWEEN CURVES
壳层法，圆盘法面积SLICE切片：
ΔV≈AΔx\Delta V\approx A \Delta xΔV≈AΔx
dv = A(x) dxV=∫A(x)dx≈∑AiΔxV = \int A(x)dx \approx \sum A_i \Delta xV=∫A(x)dx≈∑AiΔx
Solids of revolution 旋转立方体
壳层法|Disks|: dV=(πy2)dx{\color{Red} dV=(\pi y^2)dx}dV=(πy2)dx 功，平均值，概率Average value
y1+...+ynn→1b−a∫abf(x)dx\LARGE {\color{Red} \frac{y_1+...+y_n}{n}\rightarrow \frac{1}{b-a}\int_{a}^{b}f(x)dx}ny1+...+yn→b−a1∫abf(x)dxContinous average = AVE(f)
y=f(x) Δx=b−an\Delta x = \frac{b-a}{n}Δx=nb−a 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 (y1+...+yn)Δxb−a→Δx→0∫abf(x)dxb−a{\color{Red}\large \frac{(y_1+...+y_n) \Delta x}{b-a}\underset{\Delta x\rightarrow 0}{\rightarrow}\frac{\int_{a}^{b}f(x)dx}{b-a}}b−a(y1+...+yn)ΔxΔx→0→b−a∫abf(x)dx Δxb−a=1n→0(n→∞)\frac{\Delta x}{b-a} = \frac{1}{n} \rightarrow 0(n\rightarrow \infty)b−aΔx=n1→0(n→∞) WEIGHTED AVERAGE ∫abf(x)w(x)dx∫abw(x)dx=f(x){\color{Red}\large \frac{\int_{a}^{b}f(x)w(x)dx}{\int_{a}^{b}w(x)dx}=f(x)}∫abw(x)dx∫abf(x)w(x)dx=f(x)作者：KuFun人工智能
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