- #1

Addez123

- 199

- 21

- Homework Statement
- Calculate the surface of the plane

$$ax + by + c = d$$

inside the elipse

$$x^2/a^2 + y^2/b^2 = 1$$

- Relevant Equations
- Surface integrals

I start by parametarize the surface with two variables:

$$r(u,v) = (u, v, \frac {d -au -bv} c)$$

The I can get the normal vector by

$$dr/du \times dr/dv$$

What limits should I use to integrate this only within the elipse?

I could redo the whole thing and try write r(u, v) as u being the radius percentage and v being the angle such that

$$r(u, v) = (uacos(v), ubsin(v), (d - au^2cos(v) - ub^2sin(v))/c)$$

$$u: 0 \rightarrow 1, v: 0 \rightarrow 2\pi$$

But good luck calculating the cross product, even worse: the absolute value of the cross product.

It's a billion numbers, I am certain that is not the correct way to solve it.

$$r(u,v) = (u, v, \frac {d -au -bv} c)$$

The I can get the normal vector by

$$dr/du \times dr/dv$$

What limits should I use to integrate this only within the elipse?

I could redo the whole thing and try write r(u, v) as u being the radius percentage and v being the angle such that

$$r(u, v) = (uacos(v), ubsin(v), (d - au^2cos(v) - ub^2sin(v))/c)$$

$$u: 0 \rightarrow 1, v: 0 \rightarrow 2\pi$$

But good luck calculating the cross product, even worse: the absolute value of the cross product.

It's a billion numbers, I am certain that is not the correct way to solve it.

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