Question

calc iii surface integration

Answer 1

I'm really confused about this one, there's so much going on

Answer 2

\[r(r,\theta) = rcos \theta i+r \sin \theta j+r k\]
The partial derivatives are
\[r_{r}=\cos \theta i+\sin \theta j+k\]
\[r_{\theta}= -rsin \theta i+ rcos \theta j +0\]
Perform the cross product (determinant method)
\[r_{r} X r_{\theta}=\]
\[=(\sin \theta *0-rcos \theta)i-(\cos \theta*0-(-rsin \theta))j+(rcos ^{2}\theta-(-rsin ^{2} \theta))k\]
\[=-rcos \theta i - rsin \theta j +rk\]

At the point \[(2, 3/4 \pi )\], the slopes are

\[= (-2\cos (3/4 \pi), -2\sin(3/4 \pi ), 2) = (\sqrt{2}, -\sqrt{2}, 2)\]

The equation of the plane at the point \[(-\sqrt{2} , \sqrt{2} , 2)\] is
\[\sqrt{2}(x+\sqrt{2})-\sqrt{2} (y-\sqrt{2})+2(z-2)=0\]
On simplification
\[\sqrt{2} x - \sqrt{2} y +2z =0\]

Answer 3

thank you C:

would you mind helping with the next one? it's asking for the cartesian equation (same numbers)