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AREA UNDER A CURVE To
find the area under a curve, we must agree on what is desired. In figure 61,
where f(x) is equal to the constant 4 and the "curve" is the straight
line
the
area of the rectangle is found by multiplying the height times the width. Thus,
the area under the curve is
The
next problem is to find a method for determining the area under any curve,
provided that the curve is continuous. In figure 62, the area under the curve
between
points x and x +
x is
approximately
. We
consider that
x is
small and the area given is
A .
Figure 62.Area
A. This
area under the curve is nearly a rectangle. The area
A, under
the curve, would differ from the area of the rectangle by the area of the
triangle ABC if AC were a straight line. When
x becomes
smaller and smaller, the area of ABC becomes smaller at a faster rate, and ABC
finally becomes indistinguishable from a triangle. The area of this triangle
becomes negligible when
x is
sufficiently small.
Therefore,
for sufficiently small values of
x, we can
say that
Now,
if we have the curve in figure 63, the sum of all the rectangles will be
approximately equal to the area under the curve and bounded by the lines at a and b. The difference
between the actual area under the curve and the sum of the areas of the rectangles
will be the sum of the areas of the triangles above each rectangle. As
x is made smaller and smaller, the sum of the
rectangular areas will approach the value of the area under the curve. The sum
of the areas of the rectangles may be indicated by
where
(sigma) is the symbol for sum, n is the
number of rectangles,
is the area of each rectangle, and k is the
designation number of each rectangle. In the particular example just
discussed, where we have four rectangles, we would write
and
we would have only the sum of four rectangles and not the limiting area under
the curve. When
using the limit of a sum, as in equation (6.3), we are required to use
extensive algebraic techniques to find the actual area under the curve. To
this point we have been given a choice of using arithmetic and finding only an
approximation of the area under a curve or using extensive algebra to find the
actual area. We
will now use calculus to find the area under a curve fairly easily. In
figure 64, the areas under the curve, from a to b, is shown as the sum of the
areas of
and
. The
notation
, means
the area under the curve from a to c. The
Intermediate Value Theorem states that
where
f(c) in figure 64 is the value of the function at an intermediate point
between a and b.
Figure 64.Designation of limits. We
now modify figure 64 as shown in figure 65. When x=a then
We
see in figure 65 that
therefore,
the increase in area, as shown, is
Figure 65.Increments of area at fc).
Reference
to figure 65 shows
where
c is a point between a and b. Then by substitution
or
and as
x
approaches zero, we have
Now, from the definition of integration _{
} where C is the constant of integration, and _{
} but
therefore,
By solving for C, we have
and by substituting F(a) into equation (6.4), we find
If we let
then
where F(b) and F(a) are the integrals of the function of the curve at
the values b and a. The constant of integration C is omitted in equation (6.5)
because when the function of the curve at b and a is integrated, C will occur with both F(a) and F(b) and
will therefore be subtracted from itself. NOTE: The concept of the constant of integration is more
fully explained later in this chapter. EXAMPLE.
Find the area under the curve
Figure 66.Area of triangle and rectangle. in figure 66, bounded by the vertical lines at a and b
and the X axis. SOLUTION. We know that
and we find that
Then, substituting the values for a and b into
, we find
that when
and when
Then by substituting these values in
we find that
We
may verify this by considering figure 66 to be a triangle with base 4 and
height 8 sitting on a rectangle of height 1 and base 4. By known formulas, we
find the area under the curve to be 20. 