How Is the Region Defined for Rotation Around the Y-Axis?

[Lo.Lee.Ta.]

Yeah, so this might sound like a dumb question, but I'm going to ask anyway!

"Find the volume of the solid generated by revolving the described region about the given axis:

The region enclosed above by the curve y= 1 + (x^2/4), below by the x-axis, to the left by the y-axis, and to the right by the line x=3, rotated about the y-axis."


I can easily find the region the question is talking about when it says that it's bound by this or that curve.
But I'm confused here.

The curve, y= 1 + x²/4, is above the x-axis.
So how can we have another boundary that's "below the x-axis" when it also has to be above y= 1 + x²/4? ugh.

And how can the same region be to the left of the y-axis and also to the right of x=3?

This makes no sense to me. I thought there was supposed to be some region bounded by the curves that gets rotated around the y-axis!

How do I solve this sort of problem?

Thanks so much for helping! :)

 

[AlephZero]

The region enclosed above by the curve y= 1 + (x^2/4), below by the x-axis, to the left by the y-axis, and to the right by the line x=3, rotated about the y-axis."

I think your problem is understanding the English. The sentence splits up like this:

The region enclosed (bounded) above by the curve y= 1 + (x^2/4),
and enclosed below by the x-axis,
and enclosed to the left by the y-axis,
and enclosed to the right by the line x=3, rotated about the y-axis.

"Enclosed below by the x axis" means "the x-axis is the lower boundary", not "the region is below the x axis".
"Enclosed to the left by the y-axis" means "the y-axis is the left boundary", etc.

 

[Lo.Lee.Ta.]

OH, HA! I get it now! I wasn't reading it right! ;) Thanks! :D