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Numerade Educator

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Problem 8 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 = 6 - x^2 $ , $ y = 2 $ ; about the x-axis

Answer

$V=\frac{384 \pi}{5}$

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Video Transcript

were given curves and line and ratifying the volume of the solid obtained by rotating the region down by these curves. About this line were asked to sketch the region the solid and a typical disk or washer the curbs are Y equals six minus x squared Y equals two. And the line that we're rotating is about the X axis right? Yeah. Their rules on. So first I'll sketch the graph. So this first curve is a parabola. It's downward facing. It has a a vertex at 06. Who else is? Mhm. Then we have a horizontal line. Y equals two located about here. So the region that were interested is this region here in red. I remember you we want to find these endpoints here. Well we know that the why coordinate for these endpoints is too. So if you plug in to to our parabola we get two equals six minus X squared. So that x squared equals four and therefore X equals plus or minus two. So these points are negative +22 and positive to to now the X axis. Is this horizontal line here and therefore if we rotate about the X axis, we get a shape that looks something like this. Flash. You get home started, right. Try to go overseas. Like it was from her brother. I never got home started. Like it was all Flash was Flash, but I never thought it was funny. And I also thought it was like, isn't that like Cruz announcer? Yeah. Like I thought homes are was just supposed to be someone with down syndrome and your father. I mean I thought it was supposed to be literally something went down advanced. Said that some. Yeah, they would just like change like words word. It's a good job. Mhm. So this is sort of what the solid looks like. And of course if we label that's a lot of that. So a washer in red, we're in green that student green, it will look something like this. Didn't get to get a computer that is all you had. Yeah. Well we don't get a microwave and learn how to use. So this green part is our washer that's the best suited. And therefore jiffy pop our washer we see has in inner radius Of two and an outer radius. Which is, you know the other thing I know we that's a function of X. It's six minus X squared. Like so our volume is the integral using the washer method pi times the integral from X equals negative 22 X equals positive too of the outer radius squared. So six minus X squared squared minus Dina radius two squared D. X. And if you multiply this out and simplify, you eventually get after skipping a few steps. I'm not gonna do it here. This is 384 pi last season, the last season, so last episode over five