Use stoke's theorem to evaluate ∫cf⋅dr where f(x, y,z)=xi+yj+2(x2+y2)k and c is the boundary of the part of the paraboloid where z=81−x2−y2 which lies above the xy-plane and c is oriented counterclockwise when viewed from above.

Parameterize the surface (call it ) that has as its boundary by

with and .

Take the normal vector to to be

Compute the curl of . We have

Then by Stoke's theorem, the line integral is

We can verify this result by computing the line integral directly. Parameterize by

with . Then

welll its a very hard question but ill

step-by-step explanation:

first you take the 6 dollars and add 10 to it then add the 50 to that then multiply by 8 with xy and your answer will be

hope this

true !

step-by-step explanation: