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Volumes of Revolution

I am stuck on these questions. Can someone please help me
The region enclosed between the graphs of y=x and y=x^2 is denotated by R. Find the volume generated when R is rotated through 360° about
a) the x axis
b) the y axis
Have you tried anything?

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Avatar for tania62
tania62
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8
Original post by _gcx
Have you tried anything?

Spoiler



the curve and the line cross at the origin but how can you go further with only one integral?
Original post by tania62
the curve and the line cross at the origin but how can you go further with only one integral?


Where else do they cross? (what's the other solution to x2=xx^2 = x?) A sketch might help. On this sketch label R. (if your confusion lies in having to subtract two integrals, there's no problem there)
(edited 6 years ago)
Original post by tania62
I am stuck on these questions. Can someone please help me
The region enclosed between the graphs of y=x and y=x^2 is denotated by R. Find the volume generated when R is rotated through 360° about
a) the x axis
b) the y axis


If you sketch the two equations y=xy=x and y=x2y=x^2 on a single graph, you'll notice that the region enclosed by them is given by ∫01x.dx−∫01x2.dx\displaystyle \int_0^1 x .dx - \int_0^1 x^2 .dx. Now if you imagine rotating these on your graph 360 degrees around the x-axis, you'd notice that the 3D region enclosed by both is given similarly where it is (rotation of xx around x-axis)-(rotation of x2x^2 around the x-axis)

Also note how the former is simply a cone of length 2\sqrt{2} and radius 1.

Then attempt the rotation of your region around the y-axis.
Reply 5
Avatar for tania62
tania62
OP
8
Original post by _gcx
Where else do they cross? (what's the other solution to x2=xx^2 = x?) A sketch might help. On this sketch label R. (if your confusion lies in having to subtract two integrals, there's no problem there)

Thank youuu. I got my answer to be 2/15pi
Original post by tania62
Thank youuu. I got my answer to be 2/15pi


yup, have you done part b as well?

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