Question

area of the surface defined by x+y+z=1, x^2+2y^2<1

Answer 1

using calculus?

Answer 2

using green or stokes theorem in multivariable calculus

Answer 3

the way i see it is this is area of surface of inside of ellipse which lies on a plane
equation of plane:
z = 1-x-y
limits of x,y due to ellipse
-1<x<1
|y|< sqrt((1-x^2)/2)
\[\rightarrow \int\limits_{-1}^{1}\int\limits_{-\sqrt{(1-x^{2})/2}}^{\sqrt{(1-x^{2})/2}} (1-x-y) \sqrt{\frac{dz}{dx}^{2} +\frac{dz}{dy}^{2}+1} dy dx\]
dz/dx = -1
dz/dy = -1

Answer 4

i am not familiar with greens or stokes thm though so i could be doing this wrong