Lecture 47: Triple Integral

46.1 Triple Integral

46.1.1 Basic Concepts

#####Definition: Triplepoints

description: $\iiint_{\Omega}f(x,y,z)\mathrm{d}\mathbf{v}=\lim_{\lambda\to0}\sum_{k=1}^{n}f(\xi_{k},\eta_{k},\xi_{k})\Delta\nu_{k}$

Explanation

• Concept:
• A ternary function integrating into a spatial field $\Omega$;
• $\Delta\nu_{k}$ is the volume of the geometric solid in the k-th region;
• Core:
• The core of the triple integral section is the calculation of the triple integral;
• The core of triple integral calculations is to convert them into definite integrals or double integrals;

46.1.2 Basic Calculation Methods

#####Theorem: Calculationoftherectangularcoordinateoftripleintegrals

description:

1. First one, then two: $$\iiint_{\Omega}f(x,y,z)\mathrm{dv}=\iint_{D_{xy}}d\sigma\int_{z_{1}(x,y)}^{z_{2}(x,y)}f(x,y,z)dz$$

2. Two first, one later: $$\iiint_{\Omega}f(x,y,z)\mathrm{d}\mathbf{v}=\int_{c_{1}}^{c_{2}}dz\iint_{D_{c}}f(x,y,z)dxdy$$

Explanation

• First one, then two:
• Explanation:
• First, take definite integrals on z;
• Double integration on xy;
• Limiting method:
• Draw a projection surface $D_{xy}$ of the current set on the xy axis, and draw a ray upward on this projection plane passing through the geometric solid;
• The lower part of the point is the lower limit of the points $z=z_1(x,y)$, and the upper part of the part is the upper limit of the points $z=z_2(x,y)$;
• Two before one:
• Explanation:
• First, perform the double integral on xy;
• Taking definite integrals on z;
• Limiting method:
• When projecting a geometric solid onto the z axis, the upper and lower bounds of z are its maximum and minimum values projected onto the z axis;

46.2 Cylindrical Coordinates

46.2.1 Basic Concepts

#####Definition: Cylindricalcoordinates

description: $$\begin{cases}x=r\cos\theta,&\quad0\leq r<+\infty,\\y=r\sin\theta,&\quad0\leq\theta\leq2\pi,\\z=z,&\quad-\infty<z<+\infty.\end{cases}$$

Explanation

• Concept:
• r: The distance from the point to the z axis;
• z: height on the z-axis;
• angle : Limit the angle of the current line segment;
• Illustration:
• 

46.2.2 Methods for Calculating Cylindrical Coordinates

#####Theorem: Cylindricalcoordinatecalculationmethod

description:

Volume micro yuan: $dv=\rho d\rho d\theta dz$

Calculation: $$\iiint_\Omega f(x,y,z)d\nu=\iiint_{\Omega}f(\rho\cos\theta,\rho\sin\theta,z)r\operatorname{d}r\operatorname{d}\theta\operatorname{d}z$$

Supplement: Scope of application for cylindrical coordinates

• Function angle:
• If the integrand can be written as $g(x)*f(\sqrt{x^2+y^2})$ -> cylindrical coordinates are suitable;
• Regional Perspective:
• Cylindrical coordinates such as z-axis cylinders, eccentric cylinders, and cones -> are suitable for cylindrical coordinates;

46.3 Calculating Triple Integrals with Spherical Coordinates

46.3.1 Basic Concepts

#####Definition: Sphericalcoordinates

description: $$\begin{cases}x=r\sin\varphi\cos\theta,&\quad0\leq r<+\infty,\\y=r\sin\varphi\sin\theta,&\quad0\leq\varphi\leq\pi,\\z=r\cos\varphi,&\quad0\leq\theta\leq2\pi.\end{cases}$$

Explanation

• Illustration:
• 

46.3.2 Methods for Calculating Spherical Coordinates

#####Theorem: Methodsforcalculatingsphericalcoordinates

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Volume micro yuan: $dv=r^{2}\sin\varphi drd\varphi d\theta$

Calculation: $$\iiint_{\Omega}f(x,y,z)d\nu =\iiint_\Omega f (r\sin\varphi\cos\theta, r\sin\varphi\sin\theta, r\cos\varphi) r^2\sin\varphi\operatorname{d}r\operatorname{d}\varphi\operatorname{d}\theta$$

Supplement: Scope of application for spherical coordinates

• Function angle:
• If the integrand can be written as $f(\sqrt{x^2+y^2+z^2})$ -> spherical coordinates are suitable;
• Regional Perspective:
• Spherical spheres, spheres, hemispheroids, and curved pyramids centered at the origin -> suitable for spherical coordinates;

46.4 Properties of Triple Integrals

46.4.1 Parity

#####Theorem: Paritycalculationoftripleintegrals

description: $$\text{ Like a division domain }\Omega\text{ About }xoy\text{ Coordinate face-to-face symmetry },\,\text{ then }\iiint_{\Omega}f(x,y,z)\mathrm{d}V=\begin{cases}2\iiint f(x,y,z)\mathrm{d}V&f(x,y,-z)=f(x,y,z)\\0&f(x,y,-z)=-f(x,y,z)\end{cases}$$

Explanation

46.4.2 Symmetry

#####Theorem: Symmetryoftripleintegrals

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46.5 Frequently Tested Question Types

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Question Type: Calculationoftripleintegrals

PART 1: Problem-solving methods

PART 2: Typical Example Problems

PART 3: Key Points Review

Key Point: Suitable for situations where two comes first, then one

• 1. The integrand function is simply a single-variable function about z;
• 2. When using z=z to truncate the area of a geometric solid, the formula for this area is easy to calculate (for example, $x^2+y^2<1-z^2$);