Evaluate the surface integral ∫∫ f. ds for the given vector field f and the oriented surface s. in other words, find the flux of f across s. for closed surfaces, use the positive (outward) orientation.
f(x, y,z)=xi+yj+5k
s is the coundary of the region enclosed by the cylinder x^2+z^2=1 and the planes y=0 and x+y=2

Use the divergence theorem.

By the divergence theorem,

where is the region with boundary .

Compute the latter integral in cylindrical coordinates, taking

center (1,3) with radius 6

step-by-step explanation:

we must complete the square to find the center and radius of the circle.

first make sure the x and y squared terms have 1 as their coefficients. we also make sure x and y terms together.

we now create space between the x and y terms with parenthesis.

we complete the square by taking the middle terms -2x and the -6y - divide each and square them.

we add the squares to both sides.

simplify.

and write the quadratics in factored form.

the center is (h,k) or (1,3). the radius is the square root of 36 which is 6.