Chapter 8: Multiple Integrals

8.1 Double Integral

Definition: Double integral

If $f(x,y)$ is a bounded function on region D, divide region D arbitrarily into n regions: $\sigma_{1}$, $\sigma_{2}$... $\sigma_{n}$, take any point $(\xi_{i},\eta_{i})$ on each $\sigma_{n}$ and do $f(\xi,\eta_{i})\Delta\sigma_{i}$, when $\lambda\to0$:
The following expression is called a double integral: $$\lim_{\lambda\to0}\sum_{i=1}^{n}\Delta\sigma_{i}f\left(\xi_{i},\eta_{i}\right)=\int\int_{D}f(x,y)d\sigma $$

Theorem: Properties of double integrals

（1）
If on D, $f(x,y)\leq g(x,y)$, then there is an inequality:
$$\int\int_Df(x,y)d\sigma\leq\int\int_Dg(x,y)d\sigma $$
（2）
If there is a $m\leq f(x,y)\leq M$ on $D$, then:
$$mS\leq\iint_Df(x,y)\mathrm{d}\sigma\leq MS$$
（3）
The absolute value of an integral, which is less than the absolute value
$$\left|\iint_Df(x,y)\mathrm{d}\sigma\right|\leq\iint_D\left|f(x,y)\right|\mathrm{d}\sigma.$$
（4）
If $f(x,y)$ on region D is always equal to 1, then its double integral is:
$$\int\int_{D}\left(1) d\sigma=\sigma\times1\right.$$

Theorem: The mean value theorem for double integrals

Let the function $f(x,y)$ be continuous on the closed region D, S be the area of region D, then there is at least a bit of $(\xi,\eta)$ on D, such that $$\iint_Df(x,y)\mathrm{d}\sigma=f(\xi,\eta)\cdot S$$

Suitable for polar coordinates

(1) Integrands suitable for polar coordinates calculations
Formula:
$$f(x^2+y^2),f(\sqrt{x^2+y^2}),f(\frac yx),f(\frac xy)$$
Cause:
$\sqrt{x^2+y^2}$ is more complex in the Cartesian coordinate system, but represents ρ in polar coordinates
$\frac yx$ In polar coordinates, it represents an angle
(2) Integration fields suitable for polar coordinates
$$x^{2}+y^{2}\leq R^{2};\quad\quad\quad r^{2}\leq x^{2}+y^{2}\leq R^{2};\quad\quad\quad\\x^{2}+y^{2}\leq2ax;\quad\quad\quad x^{2}+y^{2}\leq2by;$$
Note:
When the center is not at the origin, $x-x_0$ can be set to $\rho\sin\theta$, and similarly, $y-y_0$
If (1) and (2) have a conflict, (1) takes precedence

Theorem: Double integral calculation based on rectangular coordinate systems

(1) First Y X: $$\int\int_D{f(x,y)d\sigma = \int_{a}^{b}[\int_{y_{1}(x)}^{y_{2}(x)}f(x,y)dy]dx}$$
Area: $\begin{aligned}\varphi_1(x)&\leq y\leq\varphi_2(x)\cdot\\a&\leq x\leq b\end{aligned}$
Where $\varphi_1(x)\leq y\leq\varphi_2(x)$ represents the range of x values, which is the function of x with respect to y, and $\varphi_1(x)$ is the actual value of y equal;
(2) First X Y: $$\iint_Df(x,y)\mathrm{d}\sigma=\int_c^ddy\int_{\psi_1(y)}^{\psi_2(y)}f(x,y)dx$$
Area: $\begin{aligned}\Psi(y)&\leq X\leq\Psi_2(y)\\c&\leq y\leq d\end{aligned}$
Setting the points limit;
When Y comes first, then X:
dy upper and lower integral limits -> Draw a ray from bottom to top, with the lower end of the ray being dy the lower limit of the integral and the upper end of the ray being the upper limit of the integral of dy;
dx upper and lower limits of integration -> Observe the graph to see the range of x values;
When X comes first, then Y:
dx upper and lower integral limits -> Draw a ray from left to right, with the left end of the ray being the dx lower limit and the upper end of the ray being the dx lower limit;
dy upper and lower limits of integration -> Observe the image and see the range for y;

Theorem: Double integral calculation based on polar coordinates

$$\text{ First }\rho\text{ Afterwards }\theta\quad\iint_Df (x, y)\mathrm{d}\sigma=\int_\alpha^\beta d\theta\int_{\varphi_1 (\theta)}^{\varphi_2 (\theta)}f (\rho\cos\theta,\rho\sin\theta)\rho d\rho$$
Area: $\begin{aligned}\varphi_1(0)&\leq p\leq\varphi_2(0)\\\alpha&\leq\theta\leq\beta.\end{aligned}$

Theorem: Parity

If the integral D relation Y axial symmetry, then the function is parity with respect to X:
$$\iint\limits_{D}f(x,y)d\sigma=\begin{cases}2\iint\limits_{D_{x\geq0}}f(x,y)\mathrm{d}\sigma;&f(-x,y)=f(x,y)\\0;&f(-x,y)=-f(x,y)\end{cases}$$
If the integral D relation X axial symmetry, then the function is parity with respect to Y:
$$\iint\limits_{D}f(x,y)d\sigma=\begin{cases}2\iint\limits_{D_{y_{z_0}}}f(x,y)\mathrm{d}\sigma&f(x,-y)=f(x,y)\\0&f(x,-y)=-f(x,y)\end{cases}$$

Theorem: Symmetry

$$\text{ If }D\text{ About }y=x\text{ Symmetric, then }\quad\iint_Df(x,y)\mathrm{d}\sigma=\iint_Df(y,x)\mathrm{d}\sigma $$
Or:
$$\int\int_{D(x,y)} f(x,y)\mathrm{d}x\mathrm{d}y=\int\int_{D(u,v)} f(u,v)\mathrm{d}u\mathrm{d}v=\int\int_{D(y,x)} f(y,x)\mathrm{d}y\mathrm{d}x$$

Definition: Triple points

$\iiint_{\Omega}f(x,y,z)\mathrm{d}\mathbf{v}=\lim_{\lambda\to0}\sum_{k=1}^{n}f(\xi_{k},\eta_{k},\xi_{k})\Delta\nu_{k}$
A ternary function integrating into a spatial field $\Omega$;
$\Delta\nu_{k}$ is the volume of the geometric solid in the k-th region;

Calculation: Cartesian coordinates

Method 1: First one, then two, $$\iiint_{\Omega}f(x,y,z)\mathrm{dv}=\iint_{D_{xy}}d\sigma\int_{z_{1}(x,y)}^{z_{2}(x,y)}f(x,y,z)dz$$
Method 2: Put two first, then one, $$\iiint_{\Omega}f(x,y,z)\mathrm{d}\mathbf{v}=\int_{c_{1}}^{c_{2}}dz\iint_{D_{c}}f(x,y,z)dxdy$$

Calculation: Cylindrical coordinates

Definition: Cylindrical coordinates
$$\begin{cases}x=r\cos\theta,&\quad0\leq r<+\infty,\\y=r\sin\theta,&\quad0\leq\theta\leq2\pi,\\z=z,&\quad-\infty<z<+\infty.\end{cases}$$
Illustration:
Calculation:
Volume microyuan: $dv=\rho d\rho d\theta dz$
$$\iiint_\Omega f(x,y,z)d\nu=\iiint_{\Omega}f(\rho\cos\theta,\rho\sin\theta,z)r\operatorname{d}r\operatorname{d}\theta\operatorname{d}z$$

Calculation: Spherical coordinates

Definition: Spherical coordinates
$$\begin{cases}x=r\sin\varphi\cos\theta,&\quad0\leq r<+\infty,\\y=r\sin\varphi\sin\theta,&\quad0\leq\varphi\leq\pi,\\z=r\cos\varphi,&\quad0\leq\theta\leq2\pi.\end{cases}$$
Illustration:
Calculation:
Volume microyuan: $dv=r^{2}\sin\varphi drd\varphi d\theta$
$$\iiint_{\Omega}f(x,y,z)d\nu =\iiint_\Omega f (r\sin\varphi\cos\theta, r\sin\varphi\sin\theta, r\cos\varphi) r^2\sin\varphi\operatorname{d}r\operatorname{d}\varphi\operatorname{d}\theta$$
Applicable to: The integrand can be written as $f(\sqrt{x^2+y^2+z^2})$ or as a sphere, sphere, hemispherical, or curved cone centered at the origin -> spherical coordinates are suitable