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'example of integration with respect to surface area of the paraboloid'
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| Title of object: |
example of integration with respect to surface area of the paraboloid |
| Canonical Name: |
Example7OfIntegrationWithRespectToSurfaceArea |
| Type: |
Example |
| Created on: |
2005-01-27 16:35:41 |
| Modified on: |
2006-06-14 11:49:19 |
| Classification: |
msc:28A75 |
Revision comment (for changes between this and next version):
| changes for correction #8236 |
Preamble:
Content:
In this example we examine the paraboloid given by the equation $z = x^2 + 3 y^2$. We have
$$\sqrt{1 + \left( \frac{\partial g}{\partial x} \right)^{\!2} + \left( \frac{\partial g}{\partial y} \right)^{\!2}}
= \sqrt{1 + \left( 2 x \right)^2 + \left( 6 y \right)^2}
= \sqrt{1 + 4 x^2 + 36 y^2 }$$
and hence
$$\int_S f(x,y) \, d^2 A = \int f(x,y) \sqrt{ 1 + 4 x^2 + 36 y^2 } \, dx \, dy.$$
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