Integrating the formula for the sum of the first $n$ natural numbers

We have Faulhaber's formula:

$$\sum_{k=1}^n k^p = \frac1{p+1}\sum_{j=0}^p (-1)^j\binom{p+1}jB_jn^{p+1-j},~\mbox{where}~B_1=-\frac12$$

$$\implies f_p(x)=\frac1{p+1}\sum_{j=0}^p(-1)^j\binom{p+1}jB_jx^{p+1-j}$$

We integrate the RHS from $-1$ to $0$ to get

$$I_p=\int_{-1}^0f_p(x)~\mathrm dx=\frac{(-1)^p}{p+1}\sum_{j=0}^p\binom{p+1}j\frac{B_j}{p+2-j}$$

Using the recursive definition of the Bernoulli numbers,

$$I_p=(-1)^p\frac{B_{p+1}}{p+1}=-\frac{B_{p+1}}{p+1}$$

Using the well known relation $B_p=-p\zeta(1-p)$, we get

$$I_p=\zeta(-p)$$

So no coincidence here!

Funny that you should use that method to demonstrate that the sum of all natural numbers is $-\frac{1}{12}$. I've seen the same claim demonstrated in a completely different way.

In case you aren't familiar with it, the Riemann Zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty n^{-s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+...$$ Your infinite sum of positive integers is given by $\zeta(-1)$.

The Riemann zeta function satisfies the reflection formula $$\zeta(1-s)=2(2\pi)^{-s}\cos\big(\frac{\pi s}{2}\big)\Gamma(s)\zeta(s)$$ Which suggests the same thing, if you evaluate it at $s=2$: $$\zeta(1-2)=2(2\pi)^{-2}\cos\big(\frac{2\pi}{2}\big)\Gamma(2)\zeta(2)$$ $$\zeta(-1)=\frac{1}{2\pi^2}\cos(\pi)\Gamma(2)\zeta(2)$$ $$\zeta(-1)=\frac{1}{2\pi^2}\cdot-1\cdot 1! \cdot \frac{\pi^2}{6}$$ $$\zeta(-1)=-\frac{1}{12}$$

However, I am somewhat inclined to think that this is merely a coincidence, as strange as that may sound. This is because the other "strange" values that can be obtained using the $\zeta$ function, such as $$\zeta(0)=-\frac{1}{2}$$ $$\zeta(-2)=0$$ are not "confirmed" or obtained from your integral in any analogous way. I would advise you to look for some ways to produce these values from your integral so that you (and any more potential answerers) will have more to work with.

Just because the integral of your function between its two zeroes is equal to this magical number does not mean that there is any sort of connection. For example, consider the formula for the sum of the first $n$ square numbers: $$\frac{n(n+1)(2n+1)}{6}$$ This has zeroes at $0, -\frac{1}{2},$ and $1$. When I take the integral from $-1$ to $0$ of this, I get $$\int_{-1}^0 \frac{n(n+1)(2n+1)}{6}dn=0$$ Which is equal to our strange value for $\zeta(-2)$, which also is a sum of squares. But wait, there's more!

$\zeta(0)$ is the sum of $1+1+1+...$. The partial sum for this would be the sum of $1$, $n$ times, and so this sum is given by $n$. And guess what? $$\int_{-1}^0 n dn=-\frac{1}{2}$$ Which also agrees with our strange value for $\zeta(0)$.

And this just keeps working! $\zeta(-3)$ is, supposedly, $\frac{1}{120}$. The sum of the first $n$ cubes has the formula $$\frac{n^2(n+1)^2}{4}$$ and guess what? $$\int_{-1}^0 \frac{n^2(n+1)^2}{4} dn=\frac{1}{120}$$

Now I've given you lots of examples to work with, and perhaps to draft a proof that the "weird" values of $\zeta(-s)$ is equal to the integral, from $-1$ to $0$, of the formula for the sum of the first $n$ $s$th powers. And hopefully this answer was helpful enough to you to be deserving of that hefty $50$-point bounty... ahem...