Calculate ∫∫x^2 dS of Triangle

[bugatti79]

Homework Statement

Calculate ∫∫x^2 dS where S is the triangle with corners (1,1,0) (0,1,0) and (0,0,1)

The triangle is the graph of g(x,y)=1-x-y

One integral can be \displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-x}x^2dy)dx]

I calculate the other one to be

\displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-y}(1-y)^2dx)dy]

but I don't get the same answer...can some one point it out?

Thanks

 

[Mark44]

Homework Statement

Calculate ∫∫x^2 dS where S is the triangle with corners (1,1,0) (0,1,0) and (0,0,1)

The triangle is the graph of g(x,y)=1-x-y

One integral can be \displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-x}x^2dy)dx]

I calculate the other one to be

\displaystyle \sqrt3 [\int_{0}^{1}(\int_{0}^{1-y}(1-y)^2dx)dy]

but I don't get the same answer...can some one point it out?

Thanks

The integrand, x², should not change when you change the order of integration.

 

[bugatti79]

The integrand, x², should not change when you change the order of integration.

OK, thanks. I am wondering did I come across situations where one does change the integrand or perhaps I am confusing it with changing the limits when there is a u substitution involved etc?

thanks

 

[SammyS]

I'm pretty sure the triangle is in the plane, y+z = 1 .