积分的运用¶
平面曲线弧长¶
直角坐标¶
\[
S(l)=\int_{a}^{b}\sqrt{1+f'^{2}(x)}dx\quad (a<b)
\]
参数方程¶
\[
S(l)=\int_{\alpha}^{\beta}\sqrt{x'^{2}(t)+y'^{2}(t)}dt\quad (\alpha<\beta)
\]
极坐标方程¶
\[
S(l)=\int_{\alpha}^{\beta}\sqrt{r^{2}(\theta)+r'^{2}(\theta)}d\theta\quad (\alpha<\beta)
\]
平面图形面积¶
直角坐标¶
\[
A=\int_{a}^{b}f(x)dx
\]
参数方程¶
\[
A=\int_{a}^{b}f(x)dx\stackrel{\text{x=x(t)}}{=}\int_{\alpha}^{\beta}y(t)dx(t)=\int_{\alpha}^{\beta}y(t)x'(t)dt
\]
极坐标方程¶
\[
A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}(\theta)d\theta
\]
旋转体体积¶
薄片法¶
垂直\(x\)轴的平面截面面积\(S(x)\)，则\(\Omega\)的体积
\[
V=\int_{a}^{b}S(x)dx
\]
曲线\(y=f(x)\)与\(x=a\)，\(x=b\)及\(x\)轴所围图形绕\(x\)轴旋转体体积
\[
V_{x}=\pi\int_{a}^{b}f^{2}(x)dx
\]
薄壳法¶
曲线\(y=f(x)\)与\(x=a\)，\(x=b\)及\(x\)轴所围图形绕\(y\)轴旋转体体积
\[
V=2\pi\int_{a}^{b}xydx=2\pi\int_{a}^{b}xf(x)dx
\]
旋转体侧面积¶
直角坐标¶
绕\(x\)轴旋转所得曲面面积
\[
A_{x}=2\pi\int_{a}^{b}f(x)\sqrt{1+f'^{2}(x)}dx
\]
绕\(y\)轴旋转所得曲面面积
\[
A_{x}=2\pi\int_{a}^{b}x\sqrt{1+f'^{2}(x)}dx
\]
参数方程¶
绕\(x\)轴旋转所得曲面面积
\[
A_{x}=2\pi\int_{a}^{b}y(t)\sqrt{x'^{2}(t)+y'^{2}(t)}dt
\]
绕\(y\)轴旋转所得曲面面积
\[
A_{y}=2\pi\int_{a}^{b}x(t)\sqrt{x'^{2}(t)+y'^{2}(t)}dt
\]
极坐标方程¶
绕极轴旋转所得曲面面积
\[
A_{x}=2\pi\int_{\alpha}^{\beta}r(\theta)\sin\theta\sqrt{r'^{2}(\theta)+r'^{2}(\theta)}d\theta
\]