What is the relationship between the integral of a function and the area under the graph of that function? Not a homework question.
In: 1
Let’s say you are in a car, and you plot your distance traveled using the value on the odometer on the y axis and time on the x axis. Starting at 0 miles and 0 hours, let’s say you ride for an hour and go a mile. The distance traveled can be plotted by the function y=x, which shows a line the goes up with a slope of 1.
Since you went 1 mile in one hour, the speedometer would have shown 1 mile/hour the whole time. The slope of the distance graph would be the derivative of the distance graph, which would match what the speedometer read at those same times. The speed can be plotted by the function y=1, which is simply a horizontal line crossing the y axis at 1.
Now, just like how y=1 is the derivative of y=x, y=x is the integral of y=1, so for the next part we will consider what the area under the speed graph means for the distance traveled graph.
If I’m traveling 1 mile in an hour, the distance traveled would increase on the first graph by 1 mile. in the second graph, from time 0 to time 1, the area under the curve is a simple square with height 1 and width 1. the area is 1*1, and so that is the distance traveled from time 0 to time 1.
To further solidify this, if you want to know how far you traveled in a half hour using the speed graph, you’d just have to look at the area under the portion of the graph from 0 hours to .5 hours, which would be 1*.5, or .5 miles, which should make sense. The area is a rectangle with height 1 and width .5. in the same time period (0-.5 hours), the distance traveled graph increased by .5 miles.
That is how i intuit that the integral is the area under the curve.
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