Hi -

I attach a drawing of the problem as I understand it, with three axes:

,

and

.

The values of

go from

to

. At

, the parabola (which lies in a plane parallel to the

plane) has equation

. This parabola will have values of

from

to

, and values of

from

to

.

What you need to do, then, to find the volume of the solid enclosed by all these parabolas, the

plane and the plane

, is:

- Find the area of the typical parabola shown. (Do this in the usual way with an integral, whose limits are to .)
- Replace by in your formula for .
- Now imagine increasing by an amount . As it does so, the volume 'swept out' is approximately . So the total volume will be . So, replace by your formula in terms of , and then work out the integral.

Have I given you enough to go on? Let me know if you want me to check your working.

Grandad