# 6basic Integration & Applicationsap Calculus

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)** ?

## Slices

We could calculate the function at a few points and |

We can make |

And as the slices We now write |

## That is a lot of adding up!

Basic Integration. Powered by Create your own unique website with customizable templates. Home Math 310.

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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Like here:

### Example: What is an integral of 2x?

We know that the derivative of x^{2} is 2x ...

... so an integral of 2x is x^{2}

You will see more examples later.

## Notation

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:

## Plus C

We wrote the answer as x^{2} but why + C ?

It is the 'Constant of Integration'. It is there because of **all the functions whose derivative is 2x**:

The derivative of **x ^{2}+4** is

**2x**, and the derivative of

**x**is also

^{2}+99**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 **x ^{2}**

Derivative: If the tank volume increases by **x ^{2}**, 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
**x**liters^{2}= 3^{2}= 9

And after 4 minutes (**x=4**):

- the flow rate has reached
**2x = 2×4 = 8**liters/min, - and the volume has reached
**x**liters^{2}= 4^{2}= 16

We can do the reverse, too:

Imagine you don't know the flow rate.

You only know the volume is increasing by **x ^{2}**.

We can go in reverse (using the derivative, which gives us the slope) and find that the flow rate is **2x**.

### Example:

- 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)
- etc

So Integral and Derivative |

We can write that down this way:

The integral of the flow rate | And the slope of the volume increase | And hey, we even get a nice explanation of that 'C' value ... maybe the tank already has water in it! - The flow still increases the volume by the same amount
- And the increase in volume can give us back the flow rate.
Which teaches us to always add '+ C'. ## Other functionsWell, we have played with If we are lucky enough to find the function on the ## Example: what is ∫cos(x) dx ?From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so: ∫cos(x) dx = sin(x) + C But a lot of this 'reversing' has already been done (see Rules of Integration). ## Example: What is ∫x |

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 an*antiderivative* 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.

### Definition

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

### Definition

Let f be a function defined on a closed interval [a, b]. Given any n we let Δx=(b-a)/n, and we let x_{k} = a + k Δx. For each k, select *sample points*x_{k}* ∈[x_{k-1}, x_{k}]. 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 x_{k}*.

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.

The sum:

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.