# Double integration

• December 10th 2008, 08:54 AM
mitch_nufc
Double integration
find the volume of the solid lying under the surface z=x^2+xy+y^2 and lying over the the region in the first quadrant bounded by the lines x=0, x=y and x+y=2 [Hint:you may find it easier taking a vertical slice] help please, full working would be fantastic. I had a try but got a weird answer so please help, thanks again.
• December 10th 2008, 09:39 AM
Jhevon
Quote:

Originally Posted by mitch_nufc
find the volume of the solid lying under the surface z=x^2+xy+y^2 and lying over the the region in the first quadrant bounded by the lines x=0, x=y and x+y=2 [Hint:you may find it easier taking a vertical slice] help please, full working would be fantastic. I had a try but got a weird answer so please help, thanks again.

did you sketch a diagram of the region the problem is talking about? that is, in the xy-plane?

note that $0 \le x \le 1$, $x \le y \le 2 - x$, and $0 \le z \le x^2 + xy + y^2$

so that the volume is given by:

$V = \int_0^1 \int_x^{2 - x} z~dy~dx = \int_0^1 \int_x^{2 - x} (x^2 + xy + y^2 )~dy~dx$

remember, when integrating with respect to y, treat x as a constant
• December 10th 2008, 09:50 AM
mitch_nufc
Thanks very much :), could you have a look at my other thread entitled double-angle formulas using complex numbers etc, you seem bright :D
• December 10th 2008, 09:58 AM
Jhevon
Quote:

Originally Posted by mitch_nufc
Thanks very much :), could you have a look at my other thread entitled double-angle formulas using complex numbers etc, you seem bright :D

i would, but i have to leave now.

just start with the left hand side of each equation and replace the powers in the complex expression with what is being sined or cosined. then expand and simplify to get the other side