Surface Area of a part of a plane inside an ellipsoid.

[Raziel2701]

Homework Statement

Find the surface area of that part of the plane 9x+10y+z=6 that lies inside the elliptic cylinder [tex] \frac{x^2}{25} +\frac{y^2}{100} =1[/tex]



2. The attempt at a solution

Once again I was just told that the surface area would be equal to the double integral of the area of the ellipse times the normal vector of the plane. Which gives me the correct answer being [tex]50pi\sqrt{182}[/tex] but I have no clue how this was obtained. I'm looking at my book for answers, for equivalencies in Stokes' Theorem that would indicate this but I can't find anything.

Is this because to calculate a surface integral, we must approximate the patch area of S and in this case we can actually find the area rather than using the cross product of the partials of the vector?

 

[LCKurtz]

If you parameterize the plane as

R(x,y) = <x, y, 6-9x-10y>

and use the formula dS = |R_x X R_y|dx dy

you get [itex]dS = \sqrt{182}\, dxdy[/itex], and the area becomes

[tex]\int\int_A (1)\sqrt{182}\, dxdy[/tex]

which is [itex]\sqrt{182}Area(A)[/itex] and, of course, the area of the ellipse is [itex]\pi(5)(10)[/itex]