
Triple Integration
Find volume of solid region in first octant by triple integration:
Solid is bounded by elliptic cylinder 2x²+y²=1 and the plane y+z=1
So far I've gotten to:
$\displaystyle \int\nolimits_{0}^{\frac{1}{\sqrt {2} }} {dx}\int\nolimits_{0}^{\sqrt {12x^{2}} } {dy}\int\nolimits_{0}^{1y} {dz}$
Is this on the right track?

Looks good so far. To avoid confusion, I would probably write the integral as
$\displaystyle \int_{0}^{\frac{1}{\sqrt {2} }} \int_{0}^{\sqrt {12x^{2}} } \int_{0}^{1y} dz\,dy\,dx.$