The Proof is Trivial!

x

Very good but it would be perfect if you fixed the typos for clarity!

Really Felix?

Solution to Problem 38:

Here's an additional problem:

Problem 27 **/*** (the ** rating very loosely)

Let $a,b,c \in \mathbb{R}$. Determine

$I = \displaystyle\quad \iiint\limits_{x^2+y^2+z^2 \leq 1} \cos(ax+by+cz) \ \, \mathrm{d} x\,\mathrm{d} y\,\mathrm{d}z$

Also, show that your result is consistent with the fact the volume of the unit sphere is $4 \pi/3$

Can you give me a hint of how to go about evaluating this integral, because whichever coordinate system I use, I eventually run into something nasty that I don't know what to do with. It's been driving me mad...

Part 1:

If I try cartesian/cylindrical coordinates, then:


So then I tried spherical coordinates, and:



Where am I going wrong?

Part 2:

Problem 36: */**

A particle is projected from the top of a plane inclined at an angle $\phi$ to the horizontal. It is projected down the plane. Prove that; if the particle is to attain it's maximum range, the angle of projection $\theta$ must satisfy:

$\theta = \dfrac{\pi}{4} - \dfrac{\phi}{2}$

Is $\theta$ measured from the horizontal or from the plane?

Haha, it is a bit of a horrible question!

Hint:

Problem 40*

Find all functions $f:\mathbb{R}\to \mathbb{R}$ which satisfy $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$ for all real numbers $x$ and $y$.

Problem 36: */**

A particle is projected from the top of a plane inclined at an angle $\phi$ to the horizontal. It is projected down the plane. Prove that; if the particle is to attain it's maximum range, the angle of projection $\theta$ from the horizontal must satisfy:

$\theta = \dfrac{\pi}{4} - \dfrac{\phi}{2}$