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Problem 231 Medium Difficulty

Find the volume of the solid generated when the region bounded by $y=1 / \sqrt{x(3-x)}, \quad y=0, \quad x=1$ and $x=2$ is revolved about the $x$ -axis.

Answer

$V=\frac{\pi}{3} \ln 4$ units $^{3}$

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Video Transcript

so the volume is generated. Volume generated is given by the Formula A B uh, you know, every squared of X minus G sweat of X and then DX. Right. Okay. So, um, you know, that is just gonna be pi. What is the no limit is one upper limit is to use who and then f is given as one over X one. It was great. Deal of experience. Three one of the square root off eggs into three minus x. Okay, So the squared here means that a square root is gonna go away. So she's gonna be won over one of her ex three minus X. Okay. And then this is Oh, zero. Right. So did you have X parties? Hero? So you just have the X right here. Okay. So what is the integral off? This guy right here? I'm gonna use partial fractions once again. Right, Because the rational function, one of the ways to go about it is partial fraction. So we're just gonna take the rational function and make it a over the 1st 1 then plus B over a 2nd 1 right? And then we can I just do the LCD and then group equate the new Raiders right with doing this many times, You gonna let X equal zero when X equals zero, Then we just have 38 Right? And then a is 1/3 and over. Let X equals one. And what is happening then? X equals three because we will eliminate a So when X equals three, you can see that this one is gonna be zero. So Ace eliminated. So this is gonna be three b, right? And therefore me is also one of three. So this interval right here becomes high. 12 Okay, uh, 1/3, X plus 1/3, three minus six. Right. So once you have this one, then the integral and do it in a different color, and the integral becomes I. This one becomes 1/3 natural Law X. And then this one becomes negative. 1/3. Not to look three minus x. And then we put one and two here. Right? So when you do our parliament minus lower limits, right? What do you have? You have I over three, right, Natural lago four. And since it's a volume that its units cute rice or the volume is given by this right here. Right can do. A lot of the women is the one of me. Just put two on every C X and that minus you Put one. Whenever you see X, you arrive with this answer. Right. So this is the volume of the volume generated, Okay.