Question

The region bounded by y=2x^2, y=0, x=0 and x=2 , is rotated around the -axis. Find the volume.

Where do I even begin with this?

Answer 1

The area of this region, ignoring the rotation initially, is
\[\int_0^2 2x^2 \, \, dx\]

Answer 2

Now, you omitted the axis to rotate around in your question.

Answer 3

If it's the \(x\) axis, then you want to find the area of each circle with radius \(y\), so the volume would be \( \pi \int_0^2 (2x^2)^2 \, \, dx \).

Answer 4

whoops sorry, yes its the x axis

Answer 5

I get how you got the number in the first comment. but how did you get from that to the next number?

Answer 6

You're finding the volume of this (poorly drawn) solid.

Answer 7

The radius of each of these circles is the y value.

Answer 8

So you want the sum of all of these circles' areas.

Answer 9

So you want \(\int_0^2 {\pi}r^2 \, \, dx\)

Answer 10

And since \(r=y=2x^2\) you get that integral as the answer.

Answer 11

oohhh that makes sense, I get it now. thanks! :)

Answer 12

haha it's doge! :)
nice picture