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- Multiple Integral Applications

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Get Started Now- Intro Lesson: a6:50
- Intro Lesson: b4:57
- Intro Lesson: c11:42
- Lesson: 111:43
- Lesson: 26:25
- Lesson: 33:33
- Lesson: 44:47
- Lesson: 515:35
- Lesson: 610:35

Recall that in Integral Calculus, we can change the variable $x$ to $u$ of an integral using u-substitution. In other words, we can change from

$\int f(x)dx \to \int f(u)du$

We would like to do something similar like this with double integrals.

**Transformations** is about changing from one variable to another. We will first start by transforming regions.

The Jacobian of a transformation $x=g(u,v)$ & $y=h(u,v)$ is the following:

It is the determinant of a 2 x 2 matrix.

Suppose we want to integrate the function $f(x,y)$ in the region $R$ under the transformation $x=g(u,v)$ & $y=h(u,v)$. Then the integral will now become:

$\int \int_R f(x,y)dA = \int \int_S f(g(u,v), h(u,v)) \left| \frac{d(x,y)}{d(u,v)}\right| du dv$

- Introduction
**Change in Variables Overview:**a)__Transformations__- Transformation = change one variable to another
- Similar to u-substitution in integral calculus
- $xy$-coordinate $\to$ $uv$-coordinate
- An Example of Change in Variable of Regions

b)__Jacobian of a Transformation__- Definition of Jacobian
- Determinant of a 2 x 2 matrix
- Deals with derivatives

c)__Change of Variables for a Double Integral__- All $x$'s & $y$'s become $u$'s & $v$'s
- Extra term is absolute value of Jacobian
- An Example

- 1.
**Finding the Transformations**

Suppose we have $\, R$, where $\, R \,$ is the region bounded by $\, y = x + 2, y = -x$, and $\, y = \frac{x}{2}$. Use the transformation $\, x = \frac{1}{3}u - \frac{1}{3}v \,$ and $\, y = \frac{1}{3}u + \frac{1}{3}v \,$ to determine the new region. - 2.Suppose we have $\, R$, where $\, R \,$ is the region bounded by $\, y = \frac{1}{x}, y = \frac{2}{x}, x = 2, x = 4$. Use the transformation $\, x = 2u \,$ and $\, y = \frac{v}{u} \,$ to determine the new region.
- 3.
**Finding the Jacobian**

Given that the transformations are $\, x = 2u + 4v^{2} \,$ and $\, y = u^{2} - 4v$, find the Jacobian. - 4.Given that the transformations are $\, x = u^{3}v^{5} \,$ and $\, y = \frac{u}{v}$, find the Jacobian.
- 5.
**Changing the Variables & Integrating**

Evaluate $\, \int\int_{R} x - ydA \,$ where $\,R \,$ is the region bounded by $\, \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 \,$ using the transformation $\, x = 2v, \,$ and $\, y = 3v$. - 6.Evaluate $\, \int\int_{R} 2xydA \,$ where $\,R \,$ is the region bounded by $\, xy = 2, xy = 4, x = 2, x = 4\,$ using the transformation $\, x = 2u, \,$ and $\, y = \frac{v}{u}$.

6.

Multiple Integral Applications

6.1

Change in variables

6.2

Moment and center of mass

6.3

Surface area with double integrals