Problem 39 Hard Difficulty

Use Simpson's Rule with $ n = 8 $ to estimate the volume of the solid obtained by rotating the region shown in the figure about (a) the x-axis and (b) the y-axis.

Answer

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Video Transcript

Okay, So this question wants us to find the volume of a solid of revolution in two ways. Using Simpson's ruled Approximate the intern. So the first case wants us to revolve it around the X axis so you can see that the whole graph is touching the X axis. So see equals pi times the integral from 2 to 10 of f of X quantity squared DX. So this is the integral we need to approximate for party. So it says to use an equals eight for Simpsons Rule. So that means that V is approximately high over three. Because Delta X is one times half of zero squared plus four f of one squared plus two f of two squared plus And then the pattern continues using Simpson zero all the way up until sorry. Thes limits should be changed because we're starting at two. This is 234 and this ends at 10. Sorry about that. So, for a, the volume is approximately 1 90 But again, based on how the graph is drawn, you might get anything between 1 60 and 200 depending on how you around. Now, for Part B, it wants the why access? So remember, for the y axis, since we're evolving around and access different to that of a function we have to use cylindrical shells. Sophie equals two pi times the integral from 2 to 10 of x times f of x d x and this is what it was. So we have a new integral that we have to approximate this time but and still equals eight. So our volume is approximately to pi over three times to f of two plus four times three f of three plus two times four effort for plus all the way up to 10 Ethel 10. And this time we get a vey valu of 8 20 aids. So again, anywhere between 800 and 8 50 would be acceptable based on how you decide to round things.