Surface area with double integrals

Surface area with double integrals

Lessons

Notes:

Surface Area with Double Integrals

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 1st 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 $x^{2} + z^{2} =1$.