- #1
KillaMarcilla
- 56
- 0
Yo guys, I'm stumped on how to parameterize this surface and then compute an integral over it
I'm supposed to compute \int\int_S \vec{r} over the surface formed by the x-y plane and z=4-(x^2+y^2), but I don't know to put it together and do it
(No matter how you work with x and y, z will always be zero for the flat part, and whether you use the normal or that \bigl(\frac{\delta S}{\delta u}\times\frac{\delta S}{\delta v}\bigr), the integrand will end up being zero for the bottom)
For the curved part, I tried using S: \cases{ x=r \cos(\theta), y=r \sin(\theta), z=4-r^2}, which gave me \bigl(\frac{\delta S}{\delta \theta}\times\frac{\delta S}{\delta r}\bigr) = 2 r \sin\theta\vec{i} - 2 r \cos\theta\vec{j} - r\vec{k}, and then I computed \int_{\theta=0}^{2\pi}\int_{r=0}^{2}\vec{r}(r\cos\theta,r\sin\theta,4-r^2)\cdot\bigl(2 r \sin\theta\vec{i} - 2 r \cos\theta\vec{j} - r\vec{k}\bigr)
But I came up with -8\pi rather than 24\pi.. I think the negative can come from how I arbitrarily decided to look at d theta cross dr instead of the other way around, sso I was looking at the flux in instead of the flux out, but I don't know how to explain the fact that I ended up with 4 - 8 integrated from zero to 2pi rather than 4 + 12 from zero to 2pi
I'm supposed to compute \int\int_S \vec{r} over the surface formed by the x-y plane and z=4-(x^2+y^2), but I don't know to put it together and do it
(No matter how you work with x and y, z will always be zero for the flat part, and whether you use the normal or that \bigl(\frac{\delta S}{\delta u}\times\frac{\delta S}{\delta v}\bigr), the integrand will end up being zero for the bottom)
For the curved part, I tried using S: \cases{ x=r \cos(\theta), y=r \sin(\theta), z=4-r^2}, which gave me \bigl(\frac{\delta S}{\delta \theta}\times\frac{\delta S}{\delta r}\bigr) = 2 r \sin\theta\vec{i} - 2 r \cos\theta\vec{j} - r\vec{k}, and then I computed \int_{\theta=0}^{2\pi}\int_{r=0}^{2}\vec{r}(r\cos\theta,r\sin\theta,4-r^2)\cdot\bigl(2 r \sin\theta\vec{i} - 2 r \cos\theta\vec{j} - r\vec{k}\bigr)
But I came up with -8\pi rather than 24\pi.. I think the negative can come from how I arbitrarily decided to look at d theta cross dr instead of the other way around, sso I was looking at the flux in instead of the flux out, but I don't know how to explain the fact that I ended up with 4 - 8 integrated from zero to 2pi rather than 4 + 12 from zero to 2pi
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