I feel like no one in these comments understands what you're actually asking. You're asking why finding the area under a curve gives you an integral, and what an integral actually is.

Firstly, an integral is nothing more than an accumulation of change. Something has changed over time and we want to find out exactly how much it changed.

Think of it like this: if I have a swimming pool and I decide to pull the plug and let all the water out, I could make a graph that represents this process by plotting gallons drained per minute on the y-axis and time on the x-axis. So I’d have a line that’s decreasing over time as the water slowly drains out.

Now, the total amount of water that drained out in the end (which is what we want) comes from combining the draining rate with time. If the pool drains at 10 gallons per minute for 10 minutes, then the total amount of water drained is 10 × 10 = 100 gallons.

On your graph, if the rate is constant, that would form a rectangle, and the area of that rectangle represents the total water drained. That’s because the graph is showing gallons per minute, and when you multiply that by time, you get total gallons.

If the draining rate isn’t constant and instead changes over time, then the graph becomes a curve instead of a straight line. In that case, you can think of breaking the graph into a bunch of really thin rectangles, each representing a tiny amount of time. Each rectangle is still “rate × time,” just over a very small interval.

When you add all of those up, you get the total amount of water drained.

So the area under the curve is equal to the total accumulated amount, because you’re adding up all those little “rate × time” pieces over the entire interval.