Problem 18 Hard Difficulty

The given curve is rotated about the y-axis. Find the area of the resulting surface.

$ y = \frac{1}{4} x^2 - \frac{1}{2} \ln x $ , $ 1 \le x \le 2 $

Answer

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Video Transcript

The first thing we can do is we can differentiate with respect to acts, which means that de y over d axe is gonna be one over to X minus one over two acts, which means that de acts over, do you? Why squared is going to be 1/4 x squared, minus 1/2 plus one over four x squared, Which means are Inter Girl is gonna be from wondered, too, to pie axe times the square root of what we just heard out. We're literally just putting this under the square root to make it easier to read. We can simplify this to be pi times the integral from 1 to 2, two times x squared, plus one de axe. It's at the point now we can integrate. We can integrate by using the power rule, which means we increase the expert by one and then divide by the new exponents. Then we can plug in, which means we are plugging in our upper bound minus are lower bound, which means we end up with the solution of 10 pi divided by three