定积分的应用
2022-05-27 23:13:34 0 举报AI智能生成
有关定积分应用的知识梳理
作者其他创作
大纲/内容
几何应用
平面图形的面积
<span class="equation-text" data-index="0" data-equation="A_1=" contenteditable="false"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_2=\begin{matrix} \int_{_a}^{b} |f(x)|\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_3=\begin{matrix} \int_{_a}^{b} |f(x)-g(x)|\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_4"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_a}^{b} (f(x)-g(x)）\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
对曲线r=r(θ)，α≤θ≤β,<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_5=\begin{matrix} \int_{_α}^{β} 1/2r^2(θ)\, \mathrm{d}θ \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_6=\begin{matrix} \int_{_α}^{β} 1/2[r_2^2(θ)-r_1^2(θ)]\, \mathrm{d}θ \end{matrix}"><span></span><span></span></span>
曲线的弧长
对于有向曲线弧，弧长元素<span class="equation-text" contenteditable="false" data-index="0" data-equation="ds=\sqrt{dx^2+dy^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="直角坐标系：L=ds=\sqrt{1+y^2(x)}dx=\sqrt{x^2(y)+1}"><span></span><span></span></span>dy
(1)对曲线y=f(x),a<span class="equation-text" data-index="0" data-equation="\leq" contenteditable="false"><span></span><span></span></span>x<span class="equation-text" data-index="1" data-equation="\leq" contenteditable="false"><span></span><span></span></span>b,<span class="equation-text" contenteditable="false" data-index="2" data-equation="∫_a^b"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="3" data-equation="\sqrt{1+{f^\prime(x)}^2}dx"><span></span><span></span></span> (2)对曲线x=g(y),c≤x≤d,L=<span class="equation-text" contenteditable="false" data-index="4" data-equation="∫_c^d"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="5" data-equation="\sqrt{1+g^\prime(y)^2}dy"><span></span><span></span></span>
参数方程：L=<span class="equation-text" contenteditable="false" data-index="0" data-equation="ds=\sqrt{x^2(t)+y^2(t)}dt"><span></span><span></span></span>（对参数方程<span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{cases}x=x(t) \\y=y(t) \end{cases}"><span></span><span></span></span>,α≤t≤β）
极坐标方程：L=<span class="equation-text" contenteditable="false" data-index="0" data-equation="ds=\sqrt{r^2(\theta)+r^2(\theta)}d\theta"><span></span><span></span></span>（对曲线r=r(θ）,α≤θ≤β)
空间曲线<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}x=x(t) \\y=y(t） \\z=z(t）\end{cases}"><span></span><span></span></span>，α≤t≤β，L=<span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \int_{_α}^{β} \sqrt{x^2(t)+y^2(t)+z^2(t)}\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
体积
平行截面已知的立体体积：V=<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} A（x）\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
旋转体的体积：对曲线y=f(x),a≤x≤b.
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V_x=\begin{matrix} \int_{_a}^{b} πf^2(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="V_y=\begin{matrix} \int_{_a}^{b} 2πf(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
旋转体的表面积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="曲线绕旋转轴旋转，P="><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="∫_a^b2πD(l)ds"><span></span><span></span></span>（其中D(l)表示该曲线到旋转轴l的距离，ds为弧长元素）
对曲线y=f(x),a≤x≤b,绕x轴旋转，<span class="equation-text" contenteditable="false" data-index="0" data-equation="P_x=∫_a^b2πf(x)\sqrt{1+f^2(x）}dx"><span></span><span></span></span>
对曲线x=g(y)，c≤y≤d,绕y轴旋转,<span class="equation-text" contenteditable="false" data-index="0" data-equation="P_y=∫_c^d2πg(y)\sqrt{1+g^2(y)}"><span></span><span></span></span>dy
物理应用
质心
直角坐标系
<span class="equation-text" data-index="0" data-equation="\begin{cases} \overline{x}={{∫_lxds} \over {s}}={{∫_a^bx\sqrt{1+{y^\prime}^2}dx } \over{s}}\\ \overline{y}={{∫_lyds}\over {s}}={{∫_a^by\sqrt{1+{y^\prime}^2}dx }\over{s}}\end{cases}" contenteditable="false"><span></span><span></span></span>
极坐标系
<span class="equation-text" data-index="0" data-equation="\begin{cases} \overline{x}={{∫_lxds} \over {s}}={{∫_{t_0}^{t_1}x(t)\sqrt{{x^\prime}^2(t)+{y^\prime}^2(t)}dx} \over {s}} \\ \overline{y}={{∫_lyds}\over {s}}={{∫_{t_0}^{t_1}y(t)\sqrt{{x^\prime}^2(t)+{y^\prime}^2(t)}dx} \over{s}}\end{cases}" contenteditable="false"><span></span><span></span></span>
密度不是常数，而是连续函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\overline{x}={{∫_lxρds} \over {∫_lρds}} \\\overline{y}={{∫_lyρds} \over {∫_lρds}}\end{cases}"><span></span><span></span></span>
平均值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\overline{y}="><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{matrix} \ lim_{x \to \ 0} {1\over{b-a}}(y_1+y_2+…+y_n)△x\end{matrix}"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="2" data-equation="{1 \over {b-a}}"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="3" data-equation="\begin{matrix} \ lim_{△x \to \ 0}Σ_{i=1}^ny_i△x \end{matrix}"><span></span><span></span></span>=<span class="equation-text" contenteditable="false" data-index="4" data-equation="{1 \over {b-a}}"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="5" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
功
<span class="equation-text" contenteditable="false" data-index="0" data-equation="W=\begin{matrix} \ lim_{λ \to \ 0}\end{matrix}\begin{matrix} \sum_{i=1}^n f(ζ_i)△x_i \end{matrix}=\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>
定积分的近似计算
辛普森公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{matrix} \int_{_a}^{b} f(x)\, \mathrm{d}x \end{matrix}"><span></span><span></span></span>≈<span class="equation-text" contenteditable="false" data-index="1" data-equation="{{b-a} \over {6n}}[y_0+y_{2n}+2(y_2+y_4+…+y_{2n-2})+4(y_1+y_3+…+y_{2n-1}）]"><span></span><span></span></span>
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