Business Calculus. Definite Integrals. 4.3 The Definite Integral. We have seen that area under a function’s derivative can represent total accumulated change in value of the original function. The definite integral combines the antiderivative with the idea
We have seen that area under a function’s derivative can represent
total accumulated change in value of the original function.
The definite integral combines the antiderivative with the idea
of area to find total accumulated change.
f (x) gives the height of each rectangle, and dx is the ∆x,
representing the base of each rectangle.
Using the integral symbol indicates that the number of rectangles
in the sum has become infinite.
an infinite sum of (signed) areas of
rectangles from x = a to x = b.
Although many functions that we deal with are always positive,
some are not. For example, it is possible for profit to be a
positive or negative value for a particular number of items x.
When a function is negative, we mean that its height is negative,
and the graph of the function is below the x-axis.
In the definite integral, the area of each rectangle is found by
multiplying the base ∆x (a positive number, usually) by the
height f (x), which now can be positive or negative, depending
on the position of the function.
So, the signed area of a rectangle could be positive or negative.
This signed area still represents accumulated change in value of
For the function f (x) at right,
is a negative
is a positive
is approximately 0
To evaluate a definite integral, we use the theorem:
If f (x) is a continuous function on [a, b] with antiderivative F(x), then
A new question:
find the total area between the
function and the x axis
from 0 to 10.
In this case, we are not asking for
signed area, but true area.
will give the negative value of the area between
the curve and the x axis from 0 to 6.
will give the positive value of the area between
the curve and the x axis from 6 to 10.
The total area can be found by
We are using the definite integral to represent two ideas:
signed area (area under the curve by summing areas of rectangles) and total accumulated change (in an application problem).
Both of these ideas can be written mathematically using the
represents the signed area under the curve f (x), or the total accumulated change of a function
whose derivative is f (x), both from x = a to x = b.
Note: the answer to a definite integral is always a number. We
interpret the number depending on the question.
Facts about definite integrals:
If f is not continuous, but f has no vertical asymptotes, then f can
be split into sections which are continuous, except possibly with
open holes at the endpoints.
For these functions, we can still find area under the curve f by
evaluating several definite integrals.
area in blue:
When given two curves, we can
look at a graph to see where one
curve is above the other curve.
It is possible that the two curves
could intersect, so that the ‘top’
curve could become the ‘bottom’
curve for different x values.
The height of a rectangle bounded
by two curves is found by subtracting
For the graph shown:
adds positive and negative values.
If f is a continuous function, we can find the average height of
the function over an interval [a, b]:
Average value of f over [a, b] is .
This is true whether f is positive, negative, or both.