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## Homework Statement

Solve the surface integral ##\displaystyle \iint_S z^2 \, dS##, where ##S## is the part of the paraboloid ##x=y^2+z^2## given by ##0 \le x \le 1##.

## Homework Equations

## The Attempt at a Solution

First, we make the parametrization ##x=u^2+v^2, \, y=u, \, z = v##, so let ##\vec{r}(u,v) = \langle u^2 + v^2, u, v \rangle##, then through computation ##| \vec{r}_u \times \vec{r}_v | = \sqrt{1+4u^2+4v^2}##, and so ##\displaystyle \iint_S z^2 \, dS = \displaystyle \iint_D v^2 \sqrt{1+4u^2 + 4v^2} \, dA##, where ##D = \{(u,v) \, | \, u^2 + v^2 \le 1 \}##. However, this is where I get stuck, because I want to use polar coordinates to evaluate the latter integral, but I am not sure whether to use ##v = r \sin \theta## or ##v = r \cos \theta##