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

answer: domain and range mean all possible values of x and y that could be substituted to the inequality. a solution means all possible values that make the inequality statement true. so while the range is still all real numbers, the solution is the shaded area, for which y will always be greater than the line of the function.

b

step-by-step explanation:

a difference is the result of subtraction. while each of these expressions has subtraction in it, a difference will be the result when subtraction is the last operation applied in the order of operations for the expression. this means subtraction must be outside of the parenthesis. only b has this.