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Problem 33 Hard Difficulty

Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves.

$ y = x^2 - 2x $ , $ y = \frac{x}{x^2 + 1} $

Answer

$V \approx 14.4504$

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Video Transcript

Okay, The first thing we want to do is we want to draw the diagram because they've specified to use a graft to find the intersection points of intersection points. Pretty straightforward. Just the origin with zero comma. Zero second intersection point. If you want to use does most, you would figure out it's 2.175 comma 0.38 I get out of recommend using some sort of graphing platform to do this because it can be a little tricky to do these on your owned by hand. Now we know we're gonna be slicing this way, As you can see from the blue line I've drawn Okay, The height we know is gonna be acts overact, scorn plus one minus x squared plus two acts a specified in the problem. We know the radius is going to simply be X in this context, which means now plugging into the we have two pi times the intro from 0 to 2.175 the radius. We just said that was acts times the height. This was a valued axe over ax squared, plus one minus x squared cost to axe times D of axe, which gives us 14.45 04