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Video transcript
So I stopped abruptly 12 minutes into the last video because 12 minutes was too long. I needed a break. But where we left off, we were in the home stretch. We were evaluating the third surface integral. And we were setting it up as a double integral with respect to r and theta. And we just had to set up the bounds. And we know that r takes on values between 0 and 1. And theta takes on values between 0 and 2 pi. So r, we're going to integrate with respect to r first. r takes on values between 0 and 1. And theta takes on values between 0 and 2 pi. And so now we are ready to integrate. So let's do the first part. Let's do this inside part right over here. And I'm just going to rewrite the outside part. So this is going to be equal to-- we have our square root of 2 times the integral from 0 to 2 pi of-- and we have d theta right over here. So that's the outside. This inside part right over here, we can rewrite it as if we distribute the r. It's r minus r squared cosine of theta. Now we're going to integrate with respect to r. So when we integrate with respect to r, cosine of theta is just a constant. So if you integrate this with respect to r, you get-- and I'll do it in that pink color-- if you integrate with respect to r, the antiderivative of r is r squared over 2. And the antiderivative of r squared is r cubed over 3 minus r cubed over 3 cosine theta. This is just a constant. And we're going to evaluate that from 0 to 1. So when you would evaluate it at 1, you get 1/2 minus 1/3 cosine theta. So you get 1/2-- I'll just do it right over here-- 1/2 minus 1/3 cosine theta. And then minus both of these evaluated at 0, well those are just going to be 0. 0 squared minus 0 squared times whatever. It's all going to be 0. So this business right over here just evaluates to 1/2 minus 1/3 cosine of theta. And so we get the integral. This is all going to be equal now to the square root of 2 times the integral from theta 0 to 2 pi of 1/2 minus 1/3 cosine of theta d theta. And this is equal to square root of 2 times the antiderivative of 1/2 is 1/2 theta. And the antiderivative of cosine theta is sine theta. So minus 1/3 sine theta. And we're evaluating it from 0 to 2 pi. When you evaluate these at 2 pi, you have-- let me just write it all out-- this is the home stretch. I don't want to make a careless mistake. We have square root of 2 times-- let's evaluate it at 2 pi-- 1/2 times 2 pi is pi minus 1/3 times sine of 2 pi-- well that's just going to be 0. And when you evaluate it at 0, 1/2 times 0 is 0. Sine of 0 is 0. So that all comes out to 0. So all of this business simplifies to pi. And we are done. We've evaluated surface three. It is square root of 2, or the third surface integral, or the surface integral over surface three. It is square root of 2 pi. And we are done. So this part right over here is square root 2, or root 2, times pi. So this entire surface integral that we started like 80 videos ago, evaluates to-- the orange part evaluates to 0. The blue part, 3 pi over 2. And then the magenta part is square root of 2 pi. So we're left with 3 pi over 2 plus the square root of 2 pi is the value of our original surface integral.