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- Multivariable Calculus
- Multiple Integral Applications

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Get Started Now- Intro Lesson: a1:16
- Intro Lesson: b5:17
- Lesson: 112:58
- Lesson: 217:06
- Lesson: 318:08

Suppose we want to find the surface area given by the function $f(x,y)$ from the region $D$. Then the surface area can be calculated using the following:

$S = \int \int_D \sqrt{[f_x]^2 + [f_y]^2 + 1} dA$

- Introduction
**Surface Area with Double Integrals Overview:**a)- Surface Area with a function with Region $D$
- Partial Derivatives

b)- Find the Region $D$
- Find the partial derivatives $f_x$ & $f_y$
- Calculate the Double Integral

- 1.
**Finding the Surface Area with Double Integrals**

Determine the surface area of the surface $\, x + 2y + 2z = 4$ that is in the 1^{st}octant. - 2.Determine the surface area of the surface $\, z = 2 - x^{2} - y^{2}$ that is above $z = 1 + x^{2} + y^{2}$ with $x \geq 0$ and $y \geq 0$.
- 3.Determine the surface area of the surface $\, y = 3x^{2} + 3z^{2} - 4$ that is inside the cylinder $x^{2} + z^{2} =1$.

6.

Multiple Integral Applications

6.1

Change in variables

6.2

Moment and center of mass

6.3

Surface area with double integrals