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Calculate the surface area of the surface obtained when the region enclosed by the given curves is revolved about the $x$-axis $$y=2x^2-8$$ $$y=x^2-1$$

This is a model problem for an exam and I really don't know what to do. I don't have any idea what shape will I get when the given curves are revolved around the x-axis. Can somebody help me with this problem?

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Can I consider these areas for the calculation

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  • $\begingroup$ You should first draw the curves and find where they interesect. [The shape you get before you revolve arund the x-axis will be*] a sort of inverted vee-shaped thing with curved edges, I expect. But you should draw it for yourself. Edit: [*] $\endgroup$ – Zubin Mukerjee Oct 13 '18 at 16:34
  • $\begingroup$ @J.Dane Nice problem! $\endgroup$ – Michael Rozenberg Oct 13 '18 at 16:36
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    $\begingroup$ An upside-down symmetrical version of the star trek logo! $\endgroup$ – Zubin Mukerjee Oct 13 '18 at 16:37
  • $\begingroup$ I drew it but I don't know how should I seperate the integral in order to calculate the surface because I don't think that the shape I get when I revolve the curves can be calculated with one integral $\endgroup$ – J.Dane Oct 13 '18 at 16:39
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    $\begingroup$ I want the surface area of the surface obtained. Am I saying it right because English isn't my native language? $\endgroup$ – J.Dane Oct 13 '18 at 16:49
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I won't do the whole problem but the $3$ surfaces are on the intervals $[0, 1.7320508],$

$[1.7320508, 2.6457513]\ \text{and} \ [2, 2.6457513]$ whereby symmetry about the y axis means doubling each area to get the total area.

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  • $\begingroup$ I don't understand what shape will I get after rotating the area $\endgroup$ – J.Dane Feb 11 at 20:40
  • $\begingroup$ @J.Dane see my added graphic to my answer. $\endgroup$ – Phil H Feb 11 at 22:17
  • $\begingroup$ Should there be a hole in the middle or not? $\endgroup$ – J.Dane Feb 11 at 22:22
  • $\begingroup$ Can you check out my edit? $\endgroup$ – J.Dane Feb 11 at 22:45
  • $\begingroup$ Yes, that looks correct. I added the void at the center as dashed lines. $\endgroup$ – Phil H Feb 12 at 0:16

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