HELP!! Volume of a solid w/ uniform cross sections of a parabola
I didn't know how to explain the problem in words so I made a scan of it.
Edit: Any help at all will be very appreciated.
Solid with parabolic cross-section
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.