Evaluate ∫∫s z ds, where s is the surface whose sides s1 are given by the cylinder x2 + y2 = 81, whose bottom s2 is the disk x2 + y2 ≤ 81 in the plane z = 0, and whose top s3 is the part of the plane z = 9 + x that lies above s2.

The integral of over is equal to the sum of the integrals of over .

Parameterize the surface by

with and . Take the normal vector to to be

Then the integral of over is

Parameterize by

with and . Since , the integral over is also 0.

Parameterize by

with and . The normal to is

so that the integral over is

Putting the results together, the integral of over is

answer: all the values are within because it has a pattern that repeats from -6 to 4.

good luck!

d i think

step-by-step explanation: