@JohnRennie Need help with solving an integral. In the process of trying to solve the heat equation, $\frac{\partial}{\partial t}T = \alpha\nabla^2 T$.

@psa hi

hey

I'll share the full question first...

The boundary conditions from (1) are,

I've done (1), so the eigenvalue problem is solved.

I'm getting stuck on an integral when attempting to find the coefficients for (3) though.

My work looks like this:

Assuming I've done everything correctly up to that point, I can't figure out the integral on the RHS (e.g. the integral on the last line).

I don't think I can help. Being good at evaluating integrals is mostly practice and I'm badly out of practice.

To clarify the step where I multiplied both sides by the three cosines in terms of p, q, and r, that was just so that I could get the Kronecker Δ.

Are you able to see if I was able to do everything correctly up until that point?

These days I just go to web site that evaluate integrals and let it do the hard work.

haha

I'm not really sure how I'd plug that one in. It's got some arbitrary constants.

$D$ is some number less than $L_x, L_y,$ and $L_z$

I'll try something...

Does the identity for $\cos(a)\cos(b)$ help?

That would simplify the integral to a sum of two cosines.

maybe...

it's so ugly...

that's what wolfram seems to think

that's just terrible

Try using $2\cos(a)\cos(b) = cos(a+b) + cos(a-b)$

Actually, that's what Wolfram has done, isn't it?

It's given you the sum of two sines.

yeah

it's not exactly as simple as I thought it would look though

@JohnRennie $x$ should be positive here, right? it's correct to integrate the initial condition*eigenfunction from 0 to L/2 rather than -L/2 to L/2? I'm interpreting it as the heat equation for a rectangular prism with side lengths $L_x, L_y, L_z$

that would only change it by a factor of two, but I figure that's right

I don't know. It would take me a while to read and digest the question and at 05:17 on a Sunday morning I'm not sure I have the energy.

LOL

no worries, I forgot the time there

enjoy your morning : )

:-)

Hello sir

Whenever you are available please answer what does passing through axis of rotation mean

I mean my textbook it is written that force passing through axis of rotation... And then the proof that its torque is zero

@ManasDogra Thanks