Integration is a way of adding slices to find the whole.
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Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area under the curve of a function like this:
What is the area under y = f(x) ?
We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):
We can make Δx a lot smaller and add up many small slices (answer is getting better):
And as the slices approach zero in width, the answer approaches the true answer.
We now write dx to mean the Δx slices are approaching zero in width.
That is a lot of adding up!
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But we don't have to add them up, as there is a 'shortcut'. Because ...
... finding an Integral is the reverse of finding a Derivative.
(So you should really know about Derivatives before reading more!)
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Example: What is an integral of 2x?
We know that the derivative of x2 is 2x ...
... so an integral of 2x is x2
You will see more examples later.
The symbol for 'Integral' is a stylish 'S'
After the Integral Symbol we put the function we want to find the integral of (called the Integrand),
and then finish with dx to mean the slices go in the x direction (and approach zero in width).
And here is how we write the answer:
We wrote the answer as x2 but why + C ?
It is the 'Constant of Integration'. It is there because of all the functions whose derivative is 2x:
The derivative of x2+4 is 2x, and the derivative of x2+99 is also 2x, and so on! Because the derivative of a constant is zero.
So when we reverse the operation (to find the integral) we only know 2x, but there could have been a constant of any value.
So we wrap up the idea by just writing + C at the end.
Tap and Tank
Integration is like filling a tank from a tap.
The input (before integration) is the flow rate from the tap.
Integrating the flow (adding up all the little bits of water) gives us the volume of water in the tank.
Simple Example: Constant Flow Rate
Integration: With a flow rate of 1, the tank volume increases by x
Derivative: If the tank volume increases by x, then the flow rate is 1
This shows that integrals and derivatives are opposites!
Now For An Increasing Flow Rate
Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap).
As the flow rate increases, the tank fills up faster and faster.
Integration: With a flow rate of 2x, the tank volume increases by x2
Derivative: If the tank volume increases by x2, then the flow rate must be 2x
Example: with the flow in liters per minute, and the tank starting at 0
After 3 minutes (x=3):
- the flow rate has reached 2x = 2×3 = 6 liters/min,
- and the volume has reached x2 = 32 = 9 liters
And after 4 minutes (x=4):
- the flow rate has reached 2x = 2×4 = 8 liters/min,
- and the volume has reached x2 = 42 = 16 liters
We can do the reverse, too:
Imagine you don't know the flow rate.
You only know the volume is increasing by x2.
We can go in reverse (using the derivative, which gives us the slope) and find that the flow rate is 2x.
- At 1 minute the volume is increasing at 2 liters/minute (the slope of the volume is 2)
- At 2 minutes the volume is increasing at 4 liters/minute (the slope of the volume is 4)
- At 3 minutes the volume is increasing at 6 liters/minute (a slope of 6)
So Integral and Derivative are opposites.
We can write that down this way:
The integral of the flow rate 2x tells us the volume of water:
And the slope of the volume increase x2+C gives us back the flow rate:
|Indefinite Integral||Definite Integral|
Read Definite Integrals to learn more.
1. The Indefinite Integral
Recall that if I is any interval and f is a function definedon I, then a function F on I is anantiderivative of f if F'(x) = f(x) for all x ∈ I. A corollary ofthe Mean Value Theorem tells us that if G is any other antiderivative of f on I, then there is a constant C such that G(x) = F(x) + C.
Let I be an interval and f a function defined on I.Then the indefinite integral of f is defined tobe the set of all antiderivatives of f on I. We denote this set by:
We call f the integrand and x the variable ofintegration.
Since all the antiderivatives of f differ only by a constant, it is customary to write:
where F is any particular antiderivative, and C is the constant of integration which implicitly takes on all real number values.
2. The Definite Integral
Let f be a function defined on a closed interval [a, b]. Given any n we let Δx=(b-a)/n, and we let xk = a + k Δx. For each k, select sample pointsxk* ∈[xk-1, xk]. Wedefine the definite integral of f on [a, b]to be
provided this limit exists and does not depend on the choice of sample points xk*.
If the limit depends upon the choice of sample points, or is undefined forany choice of sample points, then the integral does not exist.
We call a the lower limit of theintegral, and b the upper limit. We callf the integrand.
We also make the defintions:
The integral sign '∫' is an archaic 'S' and stands for 'sum' (as does ∑, of course).
If it exists, the definite integral gives you a number as its result. In this regard, x is a 'dummy variable' and the dx reminds us what variable we are integrating with respect to. We can just as easily replace x with t and write:
and the numerical value of the definite integral is unchanged.
Geometrically, if f(x) ≥ 0 we interpret the integral as thearea between the graph y=f(x), the x-axis, andthe lines x=a and x=b. More generally,we think of the integral as a 'signed' area, where the integral is equal tothe area above the x-axis less the area below the x-axis.
in the definition of the definite integral is called a Riemann sum; the definite integral is sometimes called a Riemann integral (to distinguish it from other more general integrals used by mathematicians). Whenever you have a limit as
For certain simple functions, you can calculate an integral directly usingthis definition. However, in general, you will want to use the fundamentaltheorem of calculus and the algebraic properties of integrals.
3. Rules of Integration
Let f and g be functions and let a, b and c be constants, and assume that for each fact all the indicated definite integrals exist. Then the following are true:
Constants can be pulled out of integrals:
The integral of the sum of two functions equals the sum of the integrals of each function:
The integral of the difference of two functions equals the difference of the integrals of each function:
The integral from a to b of a function equals the integral from a to c plus the integral from c to b:
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Note that there are no general rules for integrals of products andquotients. Such integrals can sometimes, but not always, be calculated usingsubstitution or integration by parts.
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We can also give some facts about inequalities involving definite integrals.
If f(x) ≥ 0 on [a, b] then
If f(x) ≤ g(x) on [a, b] then
The absolute value of an integral is less than the integral of the absolute value.