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Business Calculus

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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

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.

- Signed Area

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

the function.

For the function f (x) at right,

is a negative

number

is a positive

number

is approximately 0

- First Fundamental Theorem of Calculus

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

definite integral.

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.

- 4.4 More Definite Integrals

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:

- Area between curves

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

the heights.

For the graph shown:

is positive

is negative

adds positive and negative values.

- Average Value of a Function

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.