Fancy slash thing? I'm an early high school student so I haven't yet encountered the symbol I am looking for in school, but what is the symbol that looks kind of like a big " / " but with curled ends called?  Sometimes it has numbers/variables above or below it. I just need to know what it's called so I can research it on my own. I think it's used in calculus... maybe?
 A: It means the area under a graph (or, rather, between the graph and the $x$-axis).
For example, the area between the graph of $y=1/(1+x^2)$ and the $x$-axis would be written like this:$$\int_{-\infty}^\infty\frac1{1+x^2}dx$$You can graph it here. It can be shown that this is equal to $\pi$!
On the other hand:
$$\int_{-1}^1\frac1{1+x^2}dx$$
can be shown to be equal to only $\pi/2$. Here, the $-1$ and $1$ mean we only consider a portion of the area, between the vertical lines $x=-1$ and $x=1$.
From a physics standpoint, if $v(t)$ is the velocity of a particle in terms of time (which may change over time, e.g. if the particle speeds up or slows down), it can be shown that $\int_a^bv(t)dt$ is equal to the distance between where the particle is at time $t=a$ and where it is at time $t=b$. This, and results like it, make integrals incredibly useful for physics and engineering.
The most powerful tool for computing integrals is the Fundamental Theorem of Calculus, which relates integrals to something called "derivatives," which you learn more about in calculus.
A: The symbol $\int$ is known as the integral sign. It's used in calculus to denote integration:
$$\int 2x \, dx = x^2 + C$$
$$\int_0^2 x \, dx = 2$$
$$\int_\mathbb{R} \phi(x) \, dx = 1$$
As you can see, it's always followed by a function, and then a differential like $dx$. This indicates which variable is being used in the integration; for instance, $\int f(x,y) \,dx \neq \int f(x,y) \,dy$, but $\int f(x) \,dx = \int f(t) \,dt$.
Without numbers/variables, it represents the indefinite integral (which is another function); with two limits or with one set as superscripts or subscripts, it denotes the definite integral (which is a number).
You'll see multiple integral signs $\iint$ for an integral with multiple differentials,
$$\iint f(x,y) \,dy \,dx,$$
or a closed integral sign $\oint$ (not sure if that's the correct name) for an integral over a closed curve, surface, etc. Don't expect to see either of those in high school calculus.
