Alex Notes
Chapter 12 Multiple Integrals
| Page | Description | Equation |
|---|---|---|
| 976 980 |
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\[ \begin{multline*} \iint_{R} f\left(x,y\right)\,dA\\ =\iint_{R} f\left(x,y\right)\dx\dy\\ =\lim\limits_{\Delta A\rightarrow0} \sum\limits_{k=1}^{n} f\left(x_k,y_k\right)\Delta A_k \end{multline*} \] |
| 976 |
|
\[ \begin{multline*} \iint_{R} k f\left(x,y\right)\,dA\\ =k\iint_{R} f\left(x,y\right)\,dA \end{multline*} \] |
| 976 |
|
\[ \begin{multline*} \iint_{R}\left(f\left(x,y\right) \pm g\left(x,y\right)\right)\,dA\\ =\iint_{R} f\left(x,y\right)\,dA\\ \pm\iint_{R}g\left(x,y\right)\,dA \end{multline*} \] |
| 976 |
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\[ \iint_{R} f\left(x,y\right)dA\geq0\\ \text{if}\\ f\left(x,y\right)\geq0 \text{ on R} \] |
| 976 |
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\[ \iint_{R} f\left(x,y\right)\,dA \geq\iint_{R} g\left(x,y\right)\,dA\\ \text{if}\\ f\left(x,y\right) \geq g\left(x,y\right) \text{ on R} \] |
| 976 |
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\[ \begin{multline*} \iint_{R} f\left(x,y\right)\,dA\\ =\iint_{R_1} f\left(x,y\right)\,dA\\ +\iint_{R_2} f\left(x,y\right)\,dA \end{multline*} \] |
| 977 |
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\[ V=\iint_{R} f\left(x,y\right)\,dA \] |
| 978 |
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\[ \begin{multline*} \iint\limits_{R} f\left(x,y\right)\,dA\\ =\int\limits_{c}^{d}\int\limits_{a}^{b}f\left(x,y\right)\dx\dy\\ =\int\limits_{a}^{b}\int\limits_{c}^{d}f\left(x,y\right)\dy\dx \end{multline*} \] |
| 981 |
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\[ \begin{multline*} \iint\limits_{R} f\left(x,y\right)\,dA\\ =\int\limits_{c}^{d} \int\limits_{h_1\left(y\right)}^{h_2\left(y\right)} f\left(x,y\right)\dx\dy \end{multline*} \] |
| 981 |
|
\[ \begin{multline*} \iint\limits_{R} f\left(x,y\right)\,dA\\ =\int\limits_{a}^{b} \int\limits_{g_1\left(x\right)}^{g_2\left(x\right)} f\left(x,y\right)\dy\dx \end{multline*} \] |
| 987 |
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\[ A=\iint_{R}\,dA \] |
| 989 |
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\[ \text{av}\left(f\right) =\frac{1}{A}\iint_{R} f\left(x,y\right)\,dA \] |
| 991 992 |
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\[ M=\sum m_k, M=\delta(x,y)\,dA \] |
| 991 992 |
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\[ M_x=\sum{m_ky_k}, M_x=y\,\delta(x,y)\,dA \] |
| 991 992 |
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\[ M_y=\sum{m_kx_k}, M_y=x\,\delta(x,y)\,dA \] |
| 992 |
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\[ \mathrm{cm} =(\overline{x},\overline{y}) =\left(\frac{M_y}{M},\frac{M_x}{M}\right) \] |
| 995 |
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\[ I_x=y^2\delta(x,y)\,dA \] |
| 995 |
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\[ I_y=x^2\delta(x,y)\,dA \] |
| 995 |
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\[ \begin{multline*} I=I_z =I_x+I_y\\ =r^2\delta(x,y)\,dA\\ =x^2+y^2\delta(x,y)\,dA \end{multline*} \] |
| 994 |
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\[ I=\int{r^2\,dm} \] |
| 994 |
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\[ KE=\frac{1}{2}I\omega^2 \] |
| 995 |
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\[ I=I_z=I_x+I_y \] |
| 995 |
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\[ R_x=\sqrt{\frac{I_x}{M}}\\ R_y=\sqrt{\frac{I_y}{M}}\\ R_z=\sqrt{\frac{I_z}{M}}\\ I_x=MR_x^2\\ I_y=MR_y^2\\ I_z=MR_z^2 \] |
| 1001 |
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\[ \begin{multline*} \iint_{R} f\left(r,\theta\right)\,dA\\ =\lim\limits_{n\rightarrow\infty} \sum\limits_{k=1}^{n} f\left(r_k,\theta_k\right)\Delta A_k \end{multline*} \] |
| 1002 |
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\[ \begin{multline*} \iint\limits_{R} f\left(r,\theta\right)\,dA\\ =\int\limits_{\theta=\alpha}^{\theta=\beta} \int\limits_{r=g_1\left(\theta\right)}^{r=g_2\left(\theta\right)} f\left(r,\theta\right)r\dr\,d\theta \end{multline*} \] |
| 1003 |
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\[ A=\iint_{R} r\dr d\theta \] |
| 1003 |
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\[ \begin{multline*} \iint_{R} f\left(x,y\right)\dx\dy\\ =\iint_{G} f \left( r\cos{\theta}, r\sin{\theta} \right) r\dr\,d\theta \end{multline*} \] |
| 1007 |
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\[ \begin{multline*} \iiint_{D} F\left(x,y,z\right)\,dV\\ =\lim\sum\limits_{k=1}^{n} F\left(x_k,y_k,z_k\right) \Delta V_k \end{multline*} \] |
| 1008 |
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\[ \iiint_{D} kF\,dV=k\iiint_{D} F\,dV \] |
| 1008 |
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\[ \begin{multline*} \iiint_{D}\left(F\pm G\right)\,dV\\ =\iiint_{D} F\,dV\pm\iiint_{D} G\,dV \end{multline*} \] |
| 1008 |
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\[ \iiint_{D} F\,dV\geq0\\ \text{ if }\\ F\geq0\text{ on }D \] |
| 1008 |
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\[ \iiint_{D} F\,dV\geq\iiint_{D} G\,dV\\ \text{ if }\\ F\geq G\text{ on }D \] |
| 1008 |
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\[ \begin{multline*} \iiint_{D} F\,dV\\ =\iiint_{D_1} F\,dV\\ +\cdots +\iiint_{D_n} F\,dV \end{multline*} \] |
| 1008 |
|
\[ V=\iiint_{D}\,dV \] |
| 1009 |
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\[ \begin{multline*} \iiint_{D} F\left(x,y,z\right)\,dV\\ =\int\limits_{x=a}^{x=b} \int\limits_{y=g_1\left(x\right)}^{y=g_2\left(x\right)} \int\limits_{z=f_1\left(x,y\right)}^{z=f_2\left(x,y\right)} F\left(x,y,z\right)\dz\dy\dx \end{multline*} \] |
| 1014 | average value of a function over $D,$ a closed bounded region in space | \[ \text{av}\left(F\right)=\frac{1}{V}\iiint_{D} F\left(x,y,z\right)\,dV \] |
| 1019 | mass, discrete or continuous mass distribution | \[ M=\iiint_{D}\delta\left(x,y,z\right)\,dV \] |
| 1019 |
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\[ M_{xy}=\iiint_{D}z\,\delta\left(x,y,z\right)\,dV \] |
| 1019 |
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\[ M_{xz}=\iiint_{D} y\,\delta\left(x,y,z\right)\,dV \] |
| 1019 |
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\[ M_{yz}=\iiint_{D} x\,\delta\left(x,y,z\right)\,dV \] |
| 1019 |
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\[ \mathrm{cm} =(\overline{x},\overline{y},\overline{z}) = \left( \frac{M_{yz}}{M}, \frac{M_{xz}}{M}, \frac{M_{xy}}{M} \right) \] |
| 1019 |
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\[ I_x=(y^2+z^2)\delta(x,y,z)\,dV \] |
| 1019 |
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\[ I_y=(x^2+z^2)\delta(x,y,z)\,dV \] |
| 1019 |
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\[ I_z=(x^2+y^2)\delta(x,y,z)\,dV \] |
| 1019 |
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\[ I_L=r^2\delta\,dV \] |
| 1019 |
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\[ R_L=\sqrt{\frac{I_L}{M}},\, I_L=MR_L^2 \] |
| 1022 |
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\[ I_L=I_{\mathrm{cm}}+mh^2 \] |
| 1023 |
|
\[ \vect{c} =\frac{m_1\vect{c}_1 +\cdots +m_n\vect{c}_n} {m_1+\cdots+m_n} \] |
| 1024 |
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\[ \left(r,\theta,z\right)\\ x=r\cos{\theta}\\ y=r\sin{\theta}\\ z=z\\ r^2=x^2+y^2,\tan{\theta}=\frac{y}{x}\\ dV=\dz\,r\dr\,d\theta \] |
| 1028 |
|
\[ \left(\rho,\varphi,\theta\right),\\ 0\le\varphi\le\pi\\ x=\rho\sin{\varphi}\cos{\theta}\\ y=\rho\sin{\varphi}\sin{\theta}\\ z=\rho\cos{\varphi}\\ \rho=\sqrt{x^2+y^2+z^2}\\ dV=\rho^2sin{\varphi}\,d\rho\,d\varphi\,d\theta \] |
| 1028 |
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\[ r=\rho \sin{\varphi}\\ z=\rho\cos{\varphi}\\ \theta=\theta\\ \rho=\sqrt{r^2+z^2}\\ \] |
| 1038 |
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\[ \begin{align*} J\left(u,v\right) &=\frac{\partial\left(x,y\right)} {\partial\left(u,v\right)} = \left| \begin{matrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} &\frac{\partial y}{\partial v} \end{matrix} \right|\\ &=\frac{\partial x}{\partial u} \frac{\partial y}{\partial v} -\frac{\partial x}{\partial v} \frac{\partial y}{\partial u} \end{align*} \] |
| 1037 |
|
\[ \begin{multline*} \iint_{D} f\left(x,y\right)\dx\dy\\ =\iint_{G} f\left(x\left(u,v\right), y\left(u,v\right)\right) \left| \frac{\partial\left(x,y\right)} {\partial\left(u,v\right)} \right| \du\dv \end{multline*} \] |
| 1038 |
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\[ \begin{multline*} \iint_{D} f\left(x,y\right)\dx\dy\\ =\iint_{G} f\left(r\cos{\theta},r\sin{\theta}\right) r\dr\,d\theta \end{multline*} \] |
| 1042 |
|
\[ J\left(u,v,w\right) =\frac{\partial\left(x,y,z\right)} {\partial\left(u,v,w\right)} = \left| \begin{matrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} &\frac{\partial x}{\partial w}\\ \frac{\partial y}{\partial u} &\frac{\partial y}{\partial v} &\frac{\partial y}{\partial w}\\ \frac{\partial z}{\partial u} &\frac{\partial z}{\partial v} &\frac{\partial z}{\partial w}\\ \end{matrix} \right| \] |
| 1041 |
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\[ \begin{multline*} \iint_{D} F\left(x,y,z\right)\dx\dy\dz\\ =\iint_{G} H\left(u,v,w\right) \left|J\left(u,v,w\right)\right| \du\dv\dw\\ \end{multline*} \] |
| 1042 |
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\[ \begin{multline*} \iint_{D} F\left(x,y,z\right)\dx\dy\dz\\ =\iint_{G} H\left(r,\theta,z\right)r\dr\,d\theta\dz\\ \end{multline*}\\ \] |
| 1042 |
|
\[ \begin{multline*} \iint_{D} F\left(x,y,z\right)\dx\dy\dz\\ =\iint_{G}{ H\left(\rho,\varphi,\theta\right) \abs{\rho^2 \sin{\varphi}} \,d\rho\,d\theta\,d\varphi }\\ \end{multline*} \] |
Notes |
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| Page | Notes |
|---|---|
| 976 | The proof of the existence and uniqueness of the limit defining a double integral of a continuous function $f$ is given in more advanced texts. The continuity of $f$ is sufficient for the existence of the double integral but not a necessary one. |
| 983 | Procedure: how to integrate double integrals. |
| 1002 | Procedure: how to integrate in polar coordinates. |
| 1009 | Procedure: how to integrate triple integrals. |
| 1026 | Procedure: how to integrate in cylindrical coordinates. |
| 1030 | Procedure: how to integrate in spherical coordinates. |
| 978 |
|
| 996 1020 |
When the density $\delta$ is constant,
engineers may call the center of
mass
|
| 1037 | The transformation goes from $G$ to $R$ but we use them to change an integral over $R$ into an integral over $G.$ |