Verified answer

Calling the sphere the level surface

g(x, y, z) = x² + y² + z² - a²,

the outward unit normal is

n = ∇g/||∇g|| = <2x, 2y, 2z>/√(4x² + 4y² + 4z²) = (1/a) <x, y, z>.

So that

F•n = (1/a) (x² + y² + z²) = a

since x² + y² + z² = a² on the surface. So

∫∫s F•ndS = ∫∫s a dS = a(½ surface area of the sphere) = a(2πa²) = 2πa^3.

in case you ever took precalculus or another math classification which covers residences of applications, you study a thank you to make certain the area of the function. between the crimson flags to look out for is once you divide by using 0. on your expression, while x = 3, the denominator is 0. to that end, the area is each quantity beside 3. regrettably, the quantity you're attempting to plug in (x = 3) is the only quantity that would not artwork in this function. to work out this familiar hand, you0 can graph this function on a TI-80 3. in case you zoom in on the fringe of the graph at x = 3, you will see that there's a sparkling spot there! this is using the fact, as reported above, there merely isn't a cost of the expression at x = 3. you are able to say, nicely it feels like the respond must be 6, finding on the graph. this theory of what the respond "must be" is what limits are all approximately. The values of the function on the left and good of x = 3 all flow in the direction of 6 as you get closer and nearer. So we are saying the shrink as x is going to 3 is 6. So besides the reality that it is not technically the respond, 6 is your suited decision. 0 heavily isn't maximum strategies-blowing in any experience. the very suited answer is to assert that the expression is undefined at x = 3. This difficulty illustrates why 0/0 is noted as indeterminate. in this difficulty, 0/0 in a manner equals 6. the theory that 0/0 can equivalent something is somewhat the essence of calculus.