Integrals (Basics)
A speedometer holding at `20` meters a second for five seconds puts you 100 meters down the road, a plain width times height. Let the needle drift and that tidy rectangle buckles into a curved patch you have to rebuild.
- ▸Chop a rate curve into thin enough strips, add their areas, and the total lands on one exact number.
- ▸That number can be a distance or a probability; only the curve beneath it decides which.
- ▸Let the curve dip below the axis and the total subtracts, tracking net change, not raw distance.
Your speedometer sits at 20 meters a second and holds there. After five seconds, how far have you gone? You already know: distance is speed times time, 20 * 5, a hundred meters. Now picture that on a plot of speed against time. A flat line at twenty, running for five seconds, fences off a rectangle, and its area, height times width, is exactly that hundred meters. The total you covered is the area under the speed line. That quiet fact is the whole engine of this topic. But real trips do not hold one speed. Let the needle climb and dip and the line bends into a curve, and the tidy rectangle no longer fits underneath it. The distance is still the area trapped under the line; there is just no single rectangle that measures a curved patch. The rest of the topic is one question: how do you total the area under a shape that keeps changing?