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

$ y = x^3 $ , $ 0 \le x \le 2 $

$$

S_{x}=\frac{\pi}{27}(145 \sqrt{145}-1)

$$

Applications of Integration

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we know that we're gonna be using the formula the integral to pie from 0 to 2 times. Why, Diaz, which, in other words, essentially means we're gonna be having X cubed times square of one plus three x squared squared, which is nine X to the fourth D backs. Okay, now we know we can do some substitution. Over here you is one plus nine extra fourth, which means X cubed de acts is do you divide by 36? Which means we have two pi times integral from 1 to 1 45 you to the 1/2. Do you okay? Use the power method to integrate, which means increased the experiment by Juan divide by the new exponents. Now you're going to be plugging in our upper and are lower bounds. The lower bound plugged in. Just 2/3. You can see this is the upper ground plug. Done. This simplifies to pi over 27 times 1 45 times squared of 1 45 minus one