Consider the graph of a function, how can you find the area under it? Once you can find the area under a function it's only a small step to find the area of a shape, as the area between two functions.

You might not be able to directly, so you may start by looking for an approximation: approximate the area with many small rectangles and sum the areas of the rectangles. This is a finite process that results in an approximate result.

But you don't want an approximation, you want the exact value, how can you do this? If you make a drawing, you should see thinner rectangles approximate the area much better, so the idea is to take a lot of really small rectangles and add their areas, usually with a computer. This is still a finite process and the result is still an approximation, but now with the idea that you can find better and better approximation with smaller and smaller rectangles. You don't sum over infinite rectangles, that would be impossible, just a lot of them.

Sometimes this is the best we can do, if the shape is too jagged or if we don't know the shape with good enough precision, but it's usually perfectly fine for applications. You probably don't need to know the area of the part to the closest fraction of a squared inch, right? If the function is "nice enough" we can use theoretical results to find the "limit" of the process, i.e. the value that is approached by getting smaller and smaller rectangles. This could be described as summing over infinitely many infinitely small rectangles but it's not literally what we do on paper, only an intuitive explanation of what the result represents.

But how do we KNOW the result is accurate? You may want to choose your rectangles is such a way that their upper side is always below the function you're approximating, just touching it, and then do the same but make the rectangles just tough the graph of the function from above. Look at the shapes you get by merging together all the "shorter" rectangles and all the "longer" ones, you should see the first is completely included withing the area under the function and the second one completely includes it. That is, you know the area of the function is between the approximation you get with rectangles below the graph and the one you get with rectangles that reach above it. This is true regardless of the width of the rectangles you use. If the function is again nice enough (which we can usually assume, since it's super hard to construct one that isn't) we can prove theoretically that this lower and upper apprimations approach the same value as the width of the rectangles decreases infinitely. Since the true area is between them, it must also be that value. This is now an infinitely precise result, which we can often compute in finite time using some clever theorems. Even when we can't we can usually quantify a bound of how far off we might be, e.g. within a square inch, and that's good enough for applications.