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Find the exact area of the surface obtained by rotating the curve about the x-axis.

$ y = \dfrac{x^3 }{6} + \dfrac{1 }{2x} $ , $ \dfrac{1 }{2} \le x \le 1 $

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Surface Area $$=\frac{263 \pi}{256}$$

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 2

Area of a Surface of Revolution

Applications of Integration

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this question asked us to find the exact area of the serfs obtained by routine the curve about the X axis. Have they stated we're looking at between 1/2 of one, So those are bounds and because we're rotating it, we know we need to add to pie. Not we know. Our original is X cubed over six plus one over to X. This is the original expression we've been given. We know this is also gonna be multiplied by why Prime squared. So one plus 1/4 ext the fourth, minus one house plus one over four ox the fourth and again, As I stated, this is the formula listed in the textbook where you've got, like, two pi when they've got why prime squared to two pi r squared or whatever the formula is how you consider it to be. We know now we take the integral. We do this by increasing the exponent by one and then dividing by the new exponents. You can also simple for the fractions as you go along. If that makes it easier for you to understand and remember our bouncer from 1/2 to 1 and this was given in the problem. Now that we have this, we know you need to plug end. So we're plugging in upper minus lower and you're probably gonna need a calculator for this. We end up with approximately 1.3 pie or 3.227

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