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I am having a hard time evaluating this line integral. I am tempted to change my $x$ value to something like $\cos(u)$ and parameterize. Heres the question.

Find the mass of a wire lying along the first octant of the curve $C$ the intersection of the elliptic paraboloid $z = 2 − x^2 − y^2$ and parabolic cylinder $z = x^2$ between $(0, 1, 0)$ and $(1, 0, 1)$ if the density of the wire at position $(x, y, z)$ is $\rho(x, y, z) = xy$.

Choose $x = t$, $y = \sqrt{1-t^2}$ and $z = t^2$ where $0 ≤ t ≤ 1$ as a parametric equation of $C$.

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    I don't understand your parametrization: $$ z(t)=t^2\neq 1=2-t^2-(1-t^2)=2-x^2(t)-y^2(t) $$ – 2017-02-06

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Solving the system $ \begin{cases} z=2-x^2-y^2\\ z=x^2 \end{cases}$ yields $\;x^2=2-x^2-y^2\; \Rightarrow\; x^2+\frac{y^2}{2}=1.$

In other words the projection of the curve in the $xy$ plane is the ellipse parametrized by $$ \begin{cases} x(t)=\cos t\\ y(t)=\sqrt{2}\sin t \end{cases} $$ with $t\in [0,2\pi]$. So $C$ can be parametrized by $$ \begin{cases} x(t)=\cos t\\ y(t)=\sqrt{2}\sin t\\ z(t)=x^2(t)=\cos^2t \end{cases} $$

It follows that the mass equals $$ m=\int_{(0,1,0)}^{(1,0,1)} xy\; dr = \int_{t_1}^{2\pi} \sqrt{2}\sin t \cos t \sqrt{\sin^2t+2\cos^2t+4\sin^2t\cos^2t}\; dt $$ where $(x(t_1),y(t_1),z(t_1))=(0,1,0)$. Such a $t_1$ does not exist. Also, the parametrization you proposed does not satisfy the equations. Are you sure about the equations in the question?