Now, look at a general function f(x), as shown in (figure 4) .
suppose you want to find the area under the curve, between x=a and x=b —that is, the blue shaded area. using our method above, we can add up the areas of lots of skinny rectangles like the green one. each rectangle has area f(x)δx, and we want to sum these areas between x=a and x=b, letting δx get very small. mathematically, you may recognize this sum as a definite integral: ∫baf(x)dx.
the notation may look intimidating, but just keep in mind that the f(x)dx refers to the small rectangle areas, the ∫ sign stands, in a sense, for the s in “sum” (that’s actually how the symbol for integration was chosen), and the a and b at the bottom and top of ∫bagive you the start and end of the region you’re interested in. thus, the definite integral ∫baf(x)dx means the total area under the curve f(x) between x=a and x=b. for the function f(x) shown below, find the definite integral ∫41f(x)dx.
the figure shows the graph of the function f(x). x is measured from 0 to 5 on the horizontal axis. f(x) is measured from 0 to 6 on the vertical axis. at x equal to 0.5 the function jumps from 0 to 5, then it stays at this value till x equal to 4.5, and then the function drops to 0.