Lecture 013

Double Integral

Double Integral = \int \int_R f(x, y) d d = \lim_{m, n \rightarrow \infty} \sum_{i = 1}^m \sum_{j = 1}^n f(x_i, y_j) \Delta Area_{ij}

For Rectangle: \begin{align*} \int \int_R f(x, y) dA &= \int_c^d (\int_a^b f(x, y) dx) dy\\ &= \int_a^b (\int_c^d f(x, y) dy) dx \end{align*}

Average Value:

2D Example: area between y = x^2 and y = \sqrt{x} \begin{align*} R &= \{(x, y) | 0 \leq x \leq 1, x^2 \leq y \leq \sqrt[3]{x}\}\\ &= \{(x, y) | 0 \leq y \leq 1, x^2 \leq y^3 \leq \sqrt{y}\}\\ \end{align*}

\begin{align*} \int \int_R 1 dA &= \int_0^1 \int_{x^2}^{\sqrt[3]{x}} 1 dy dx\\ &= \int_0^1 (y |_{x^2}^{\sqrt[3]{x}}) dx\\ &= \int_0^1 \sqrt[3](x) - x^2 dx\\ &= \frac{5}{12}\\ \end{align*}

3D Switch Order Example: signed volume of 3xy under region between y = x^3, y = x^2+1, x = 0, x = 1 \begin{align*} D &= \{(x, y) | 0 \leq x \leq 1, x^3 \leq y \leq x^3 + 1\}\\ &= D_1 \cup D_2\\ &= \{(x, y) | 0 \leq y \leq 1, 0 \leq x \leq \sqrt[3]{y}\}\\ \cup &\{(x, y) | 1 \leq y \leq 2, \sqrt[3]{y-1} \leq x \leq 1\}\\ V &= \int_0^1 \int_{x^3}^{x^3 + 1} 3xy dy dx\\ &= \int_0^1 \int_0^{\sqrt[3]{y}} 3xy dx dy + \int_1^2 \int_{\sqrt[3]{y-1}}^1 3xy dx dy\\ \end{align*}

Double Integral in Polar Coordinates

Polar Rectangle: R = \{(r, \theta) | a \leq r \leq b, \alpha \leq \theta \leq \beta\} (a sector)

Application of Probability

Probability Density Function (pdf):

Joint Density Function: for continuous random variable x and y

X~Exp(n): \frac{1}{n}e^{-X/n}

Expected Value: E(x) = \int_R x f(x) dx if x is a continuous random variable with pdf f(x)

Triple Integrals

\iint_E f(x, y, z) dV = \lim_{l, m, n \rightarrow \infty} \sum_{i = 1}^l \sum{j = 1}^m \sum_{k=1}^n f(x_{ijk}, y_{ijk}, z_{ijk}) \delta V_{ijk}

The volume of standard n-complex is \frac{1}{n!}

u-substitution: dx = \frac{dx^3}{2x^2} integrate by part: \int f(x)g(x) dx = f(x) \int g(x) dx - \int (\int g(x) dx) \frac{d f(x)}{dx} dx

// TODO: review integrate by part

Change of Variable (coordinate system)

Planar Transformation: a function T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 (from (u, v) coordinates to (x, y) coordinates)

Jacobion: if T(u, v) = \langle{x(u, v), y(u, v)}\rangle, then \frac{\partial (x, y)}{\partial (u, v)} = \det \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}\\ \end{vmatrix}

T(x, y, z) = \langle{x(u, v, w), y(u, v, w)}\rangle, then \frac{\partial (x, y, z)}{\partial (u, v, w)} = \det \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u}\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v}\\ \frac{\partial x}{\partial w} & \frac{\partial y}{\partial w} & \frac{\partial z}{\partial w}\\ \end{vmatrix}

Then \begin{align*} &\int \int_R f(x, y) dA\\ = &\int \int_S f(x, (u, v), y(u, v)) \left\|\frac{\partial (x, y)}{\partial (u, v)}\right\| du dv\\ \end{align*}

Special Case for Cylindrical Coordinates:

\|\frac{\partial (x, y, z)}{\partial (r, \theta, z)}\| = \det \begin{vmatrix} \cos \theta & -r\sin \theta & 0\\ \sin \theta & r\sin \theta & 0\\ 0 & 0 & 1\\ \end{vmatrix} = \|r\cos^2 \theta + r \sin^2 \theta + 0\| = r

Special Case for Spherical Coordinates:

\|\frac{\partial (x, y, z)}{\partial (\rho, \theta, \varphi)}\| = \det \begin{vmatrix} \cos\theta\sin\varphi & -\sin\theta\sin\varphi\rho & \rho\cos\theta\cos\varphi\\ \sin\theta\sin\varphi & \rho\cos\theta\sin\varphi & \rho\sin\theta\cos\varphi\\ \cos\varphi & 0 & -\rho\sin\varphi\\ \end{vmatrix} = \|-\rho^2\sin\varphi\| = \rho^2\sin\varphi = r\rho

Jacobian Matrix: Best linear map approximate f at (x_0, y_0). Even though the transformation is not linear in global sense, it can be linear in local sense. (See Video)

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