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Numerade Educator

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Problem 8 Medium Difficulty

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

$ y = \sqrt{5 - x} $ , $ 3 \le x \le 5 $

Answer

$$
13 \pi / 3
$$

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Video Transcript

this question asked us to find exact area of the surface by rotating around the X axis. The first thing we know we need to do is take the derivative of the function. You can use the power rule taking into account your trick properties as well. Now they've said X is between three and five, which means with are integral, are bounds, are lower bound, three upper bound five and then we know because we're rotating around the curve. We know we have to add to pie in front. Now we know we can combine this to make it a little easier to read. We could write this as pi times the integral from 3 to 5 spurt of 21 minus four Axe de axe. No, because we know we have denominator to squirt of five minus X that we canceled out. What we know we can do is we can take out a negative pi over four. Now. What we know we have to do is we have to change the bounds. We originally had excess five and excess three. Now, remember for this question using U substitution if you is 21 minus for X, then Like I said, you need to change the bounds based on this. So we have negative four DX. The bounds are gonna be changed from one 29 and then we're using you. You was 21 minus for X to the one cough d'you We know this simplifies to negative pi over sex you to the three over to remember you get the integral by increasing the exploited by one dividing by the new exponents again, we're flipping the bounds. As I stated earlier when we did the u substitution plug in. We know, obviously the answer. Hostile, Positive. As you could see right now, this is the reason why we put the balances cause negative negative cancels out and gives us positive 13 pie over three.