Evaluate the integral of the vector field defined by $$F(x,y,z)=(x,y,z)$$ over the part of the unit sphere between $$z=\frac{\pm 1}{\sqrt{2}}$$
We know that the parameterization of the unit sphere is $$U(\varphi, \theta)=(\sin\varphi\cos \theta, \sin\varphi\sin \theta, \cos \varphi)$$ and our integral is $$\iint_\limits{S}F\cdot n\,d\sigma=\iint_\limits RF(U)\cdot\left(\frac{\partial U}{\partial \varphi}\times \frac{\partial U}{\partial \theta}\right)\,d\varphi\,d\theta$$ the details of the integrand aren't important now, the important thing is the region $R$, how can I describe it?
You want to integrate over the region $-\frac{1}{\sqrt{2}} \leq z \leq \frac{1}{\sqrt{2}}$ and $z = \cos\varphi$, so you can describe the region $R$ purely in terms of $\theta$ and $\varphi$ by restricting the values of $\varphi$ appropriately.
You can visualise $R$ as a thick band around the equator (alternatively, it's the sphere with a cap removed from the north and south poles).