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
You are a rectangle. Your area is your height times your width.
But, you already knew that. So let’s get a little more specific. We would like to know the area of a function traced from x = 0 to x = 137. As a rectangle, you recognize that your height would be y(0) and your width would be 137. A bit of multiplication and presto! The area you were looking for. Almost. Not very accurate for any mildly interesting function. What to do?
Let’s make TWO rectangles, the widths of both rectangles are now half of 137 and the height of the first rectangle is the same, but the height of the second rectangle is the value for y at at x = 137 divided by 2. Now there are two rectangles, you add their areas and presto! The area you were looking for, but now more accurate. Not accurate enough? Then make THREE rectangles using the same procedure, add their areas and presto! (Again).
Keep adding rectangles and watch the accuracy get better and better as you go from 3 rectangles to 10, to 100, to… 9,999,999,999,999,999 and off to infinity. And, at infinity, finally, the precise answer for area under the function from x = 0 to x = 137. Presto.
Latest Answers