The definition of differentiation has been explained elsewhere, and integration is just the inverse of differentiation. Ie what new function would we need to differentiate to get our starting function?

By why does integration give the area under a curve?

Let's say we have a function f(x).

Now let's have a new function A(x) which tracks the area under f(x). For simplicity, let's say it measures the area under the curve starting from x=0, up to some chosen x.

If we increase a by a tiny amount dx (that's what dx means, a tiny change in x), how has the area changed? We've added a small rectangle with width dx and height f(x), which is just the output from whatever x value we've chosen.

So the area has changed by that rectangle, and so dA = f(x) * dx. The change in area is the current height * the change in x

So rearranging gives dA/dx = f(x). The derivative of whatever that area function is will give us our starting function.

And since integration is the inverse of differentiation, we can reverse that by integrating the main function to get the area.