Take a function related to x and call it f(x). f(x) = x^2 - 7, or whatever.
Integrating that gives you the area underneath that function's graph between two points.
For instance INT(x^2 -7) dx gives 1/3x^3 - 7x (The rule for simple integration like this one is add one to the power and dive by the new power. It gets harder). The find the area under it's graph between to points on the x-axis - say, one and three - you set x equal to those numbers and take the lowest from the highest; so:
1/3(3^3)-7(3) - [1/3(1^3) - 7(1)] is the area.
The opposite of integration is differentiation. Differentiating a curve gives its gradient at each value of x. It is the opposite of integration; differentiating an integrated function will return to the orginal function. The rule here is to multiply by the power then reduce the power by one.
So for y = x^2 - 7, the gradient at point x (dy/dx) = 2x