Question

Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders y = 1 - x 2, y = x 2 - 1 and the planes x + y + z = 2, 6x + y - z + 16 = 0.

Answer 1

Start with sketching the solid/cross section.
Cylinders are easier to sketch.
Draw the trace (projection on the x-y plane. The shape stays the same for the full height.
The trace has the two parabolas meet at (-1,0) and (1,0).

Draw the two planes.
Start with
x+y+z=2
x=-1,y=0 => z=3
x=1,y=0 => z=1
Draw the intersection with the cylinder.
Similarly draw the intersection of
6x+y-z+16=0 with the cylinder.

Here's a sketch drawn:

http://img515.imageshack.us/img515/2940/1329430592.jpg

Now we will integrate x from -1 to 1 (the intersections).
y=x²+1 to y=1-x² (the two parabolas)
and
z from 2-x-y to 6x+y+16 (two planes).
namely
∫∫∫dz dy dx using the previously mentioned limits.

I got 112/3 for the volume.


In this particular problem, the volume can be calculated by integrating the area of the trace (8/3)
multiplied by the average height of the planes (intersection with the z-axis (16-2)=14 to get 112/3.