Homework 12.3 Ex #42

Find the volume of the solid contained by  and . 
Here’s what this looks like with the axes turned so the y axis is vertical (borrowed from http://local.wasp.uwa.edu.au/~pbourke/geometry/cylinders/ - which is a very interesting paper extending this idea to multiple cylinders in interesting ways.)

Observe that the cross-sections of the surfaces occur on diagonal planes and are ellipses with eccentricity (√2)/2.  If you did exercise #42 in section 6.2 (If I recall…) then you know a simple way to work this volume: observe that the cross-sections are squares with area =  so you simply integrate  from bottom to top:   

Nice and tidy.  But it does involve a human with key insight into doing the cross-sections that way.  What if you don’t see that particular innovation and you’re a 2A student working from basic principles of section 12.3 in our text you might think you need to integrate z over D = circle of radius r centered at the origin in the xy plane.  This D can be described by the inequalities .
Given one of these (x,y) points we’ll have .  Thus the height of a rectangular dA in D could be sampled to produce rectangular solids of base area dA height  . 

Below the integral is set up to integrate this function over the circle of radius r in the    

So there it is, the hard way!