Addition and subtraction are inverse operations. Multiplication and division
are inverse operations.
Integration is the inverse
operation for differentiation.
Suppose f(x) = x,
and f(x) is the derivative of the function g(x).
We know the derivative of x2 is
x. So g(x) must contain x2.
But we also know that
the derivative of a constant C is zero.
So, in general we must have g(x) = x2 + C.
And, the derivative
of g(x), is f(x).
Finding this function g(x) whose derivative is f(x) is called
integrating f(x).
We use the symbol ∫ to mean integration.
In terms of our functions f and g we write: g(x) = ∫ f(x)dx
and we would read this as "g(x) is the integral of f(x)."
The notation dx names
the independent variable, in this case x.
We understand dx to mean we are
"integrating with respect to the variable x."
Examples:
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The derivative of
g(x) = x4 is x3.
We now add the constant C and get
g(x) = x4 + C.
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The derivative of
g(x) = x5 + x2 is x4 + x.
We add the constant C and get g(x) = x5 + x2 + C.
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The derivative of
g(x) = x6 + 2x is x5 + 2.
We add the constant C and get g(x) = x6 + 2x + C.
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Now that we have an elementary understanding how derivatives and integrals
are related,
we need to pin down, mathematically, what an integral is. Before we do that
we need to know that an integral is also called an antiderivative. This name makes sense
when we think of integration as the inverse of differentiation.
A function F is called an antiderivative of the continuous function f
on the interval [a,b]
iff
F is continuous on [a,b] and
F'(x) = f(x) for all x ∈ (a, b)
A closer look at this definition tells us we must be talking about continuous functions on a closed interval.
Both functions must be continuous on the same closed interval [a,b].
And finally F'(x) and f(x) evaluate to the same value on the open interval (a,b).
We can pin down the integral to a closed interval [a,b] by specifying the interval limits
on the integral sign like so:
The limits a and b are written respectively at the bottom and the top of the integral sign.
This is the Definite Integral; definite in that the lower and upper limits are
included. a and b are the limits of integration
and we
read this notation this way: we are integrating f from a to b. (If the limits are not specified as in
the first examples, the integrals are called indefinite integrals.)
The integral sign ∫ is an elongated S. The S refers
to "Sum." We will have more to say about this later. This integral sign dates
back to the mathematician Leibniz.
The fundamental theorem of Integral Calculus states that
if F(x) is the antiderivative of the the continuous function f over
the interval [a,b] then
In this theorem we attach meaning to the limits on the definite integral;
the definite integral is the difference between the values of the antiderivative
at endpoints a and b.
Examples (using the results from the previous examples)
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g(x) = x4 + C.
So g(2) - g(0) =
24 + C - (0 + C) =
16 + C - 0 - C = 16.
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g(x) = x5 + x2 + C.
g(6) - g(-1) =
65 + 62 + C - ((-1)5 + (-1)2 + C) =
7776 + 36 + C - (-1 + 1 + C) =
7812 + C - C = 7812
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g(x) = x6 + 2x + C.
g(2) - g(1) =
26 + 2(2) + C - (16 + 2(1) + C) =
64 + 4 + C - (1 + 2 + C) =
68 + C - (3 + C) = 68 - 3 + C - C = 65
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Notice when the indefinite integral becomes the definite integral, when we evaluate the
definite integral, the constant C always subtracts to 0 in the calculation.
The antiderivative
specifies the general function which is used to calculate the definite integral. It is a conceptual error
not to include the trailing constant C in the antiderivative.
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