Problem 3 Easy Difficulty

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = \sqrt{x - 1} $ , $ y = 0 $ , $ x = 5 $ ; about the x-axis

Answer

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Video Transcript

So for this question, we want to find the volume of the solid obtained by rotating the region. Bounded by why equals the square root of X minus one y equals zero X equals five on rotating back about the X axis. So first tells us to sketch the region. And in general, this is the best way to first approach the problem. So you can first begin with the Y equals zero in X equals five. So this is the access right here. The y equals euros, just the X axis. So you don't really need to draw anything. The X equals five. You can just call this point five Miss George down. So that's the X equals five line now for the y equals screw of X minus one when X is one of y equals zero. So you know that point right off the back. And then the square root function typically looks like, um, this shape where it continues to grow but grows at a slower rate as X increases. So this is a region bounded by these three lines flash curs, and we're rotating it about the X axis. So looks like that now the uh, the second step you usually want to dio when you're finding these volumes is to find the cross sectional area. And when you notice that if you're rotating about the x axis, you're really drawing circles with the radius which is equal to y in this case so we can draw them that that cross sectional area as a function of X is equal to pi r squared, which is the area of a circle. And because our is equal to X minus one. Because at this point, at this point, right here, the radius is just the white height, which is X minus the square root of X minus one. You can plug that in and get the function A of X is equal to pi. Explains one. So now that we know that we can go ahead and start calculating the volume itself So we start off with our general, you go back, you'LL see that we'LL be integrating from one X equals one two x equals five because those are the two countries of the region. So even from one to five Oh oh, and then when we do the interval in terms of finding volume we want to put affects inside the interval and from the previous page, we knew that this was equal to the interval of one to five of pie Ex Nice one, Jax. So Pie doesn't have any relationship with the acts so we can pull that out. And now we can just use anti differentiation to find the inner world this so pious still on the outside and the anti derivative of X is just x squared over two. The anti driven of one is just X. We're evaluating them from one to five. Right now. You can just compute this next page. So first we plug in five. So five squared is twenty five twenty five over two minus five. That minus plugging one is just one over to minus one. Calculate all of that. Want to buy buy pie because we put up high from the very beginning you'LL get If you compute all this out, that equals pi times fifteen over two plus one half that the end is equal to eight pi and that is our final answer