What is the graphic
representationof the
approximation of an area
under a curveusing
integration?
It’s easier to see the approxi-
mate area under a curve using
integration by means of
graphs—it all has to do with
rectangles. The idea is to
extend lines from the ends of
the curve (here, f(x)) to the x-
axis (or y-axis, depending on
the curve); we’ll call the total
area under the curve , then
divide the entire area under
the curve into equal-width sec-
tions (x 1 , x 2 , x 3 , and so on) that
are equal to parts of (the
subregions 1 , 2 , and so on).
The next step is to figure out
the area of a rectangle if each
section was “cut off” below the
curve and then above the
curve. This creates rectangles
defined by left- and right-end
points. From the left- and
right-sums, and a few more
calculations, we can approxi-
mate the area of . This can be
seen graphically on the accom-
panying charts:
What is the definite integral?
In actuality, the area is actually determined using limits. In the function f(x), as ngets
larger, the numbers determined by left (n) and right (n) will get closer and closer to
the area . This is seen as the following notation:
Area()X = "" 33 =
limLEFT n() limRIGHT n()
nn
229MATHEMATICAL ANALYSIS
To calculate the area beneath a curve (top), you can first divide the
area into equal parts with rectangles both beneath (middle) and
above (bottom) the curve. Adjusting the width of the curves will
result in an estimate that closely approximates the actual area.