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- Multivariable Calculus
- Multiple Integral Applications

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Get Started Now- Intro Lesson: a6:50
- Intro Lesson: b4:57
- Intro Lesson: c11:42

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

6.

Multiple Integral Applications

6.1

Change in variables

6.2

Moment and center of mass

6.3

Surface area with double integrals