Multivariable calculus problem

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

I have attached the problem and solution. I don’t know how to do part b even I have looked at the solution. How to transform the original cartesian equation to the semi cylindral coordinate equation? Is there is systematical way to transform it without thinking about the sketch?

 

[andrewkirk]

The integral is a 'volume integral', meaning a function (in this case the function $z\mapsto 2z$) is integrated over a 3D volume (in this case V).

The formulas for volume integrals are set out on this wiki page. That is what they have used for the first triple integral on the last formula line in the OP. That is the integral of the function $z\mapsto 2z$ over the sector of the cone with cylindrical angle coordinate $\theta$ between $-\pi/4$ and $\pi/4$. They then take away the integral over the part of the cone above triangle with vertices (0,0,0), $(1/\sqrt2,-1/\sqrt2,0),\ (1/\sqrt2,1/\sqrt2,0)$.

However I think the formula is wrong because they need to take away another two volumes from the cone sector in the first term, to match the original volume V. Those are (1) the part of the cone sector with $y<-1/\sqrt2$ and (2) the part of the cone sector with $y>1/\sqrt2$. So the formula should consist of four triple integrals, not two. The last two can be combined and multiplied by 2, since they give the same value, but as far as I can see, they can't get away with only two triple integrals if they start with a cylindrical integral over a cone sector.

Further, given that those last two triple integrals are messy, I suspect it would be easier just to do the whole thing in Cartesian cords.

 

[LCKurtz]

I disagree with @andrewkirk: The author's second solution can be done with those two integrals, but the y limits on the last triple integral are wrong. The y limits should be -x to x. Then both methods give $\frac \pi 8 - \frac 1 6$.
And @Clara Chung: No. You should always use a sketch especially when changing coordinates, or orders of integration for that matter.