Calculate the surface integral $$\iint_S yds$$ where $S$ is the part of surface $x=2y^2+1~(y>0)$ bounded by $x = y^2+z^2,~x=2,~x=3$
I drew the graph
but still it's not obvious how to proceed. It seems I can't get surface integral well, unfortunately.
- How can I parametrize this region?
The question seeks you to evaluate $ \displaystyle \iint_S y \ ds$ where $S$ is part of the surfacec $x = 2y^2+1~(y>0)$.
Surface $S$ is a parabolic cylinder and all points on its surface can be uniquely identified if we know $x$ and $z$ coordinates OR we know $y$ and $z$ coordinates.
So parametrize surface $S$ either in terms of $(x, z)$ or in terms of $(y, z)$.
$r(y, z) = (2y^2+1, y, z), \ y \geq 0$
Having parametrized the surface, now find $|r_y \times r_z|$.
To find bounds for the integral, note that $S$ is bound between $2 \leq x \leq 3$ and paraboloid $x = y^2 + z^2$.
You can check that between $x = 2$ and $x = 3$, there is part of $S$ inside the paraboloid. At their intersection,
$x = 2y^2 + 1 = y^2 + z^2 \implies - \sqrt{y^2 + 1} \leq z \leq \sqrt{y^2+1}$
Now for surface $S$, at $x = 2, y = \frac{1}{\sqrt2}$ and at $x = 3, y = 1$.
Integral to find surface area,
$\displaystyle \iint y \ |r_y \times r_z| \ dz \ dy$
Can you take it from here?