What are indefinite integrals used for?

In integral can be thought of as an area underneath a curve.  Integrals are often used to manipulate position, velocity, and acceleration equations to estimate different situations.  If you are given an equation that represents the velocity of a golf cart driving, like so: (the ‘ in y’ represents the differential element that results when one performs a derivative on the position function f(x).)  then you can find use an integral (Anti-Derivative) to get an expression for the position of the golf cart, y(x).

The integral of a function is represented like so: and it can be thought of as a sum of areas like so:

Here the integral is performed on the function from point a to point b, which would make it a definite integral because the bounds are defined.  Our indefinite integral is the same procedure, except missing the bounds, which makes indefinite integral operation require a little twist.

We all know that the derivative of a constant is zero.  For instance, the derivative of 5 is equal to zero.  Once performing this derivative, we should still be able to perform the anti-derivative on this new function (zero) to obtain the original equation (5).  But how will we know what number permeates from performing an anti-derivative of zero.  An indefinite integral is called indefinite because the bounds are not defined on the integration, like so:  .  When we perform our indefinite integral we represent this long-lost constant by the letter ‘C’.

Before we integrate our golf cart velocity equation, lets go ahead and look at the laws of integration:

The integral can also be split up into separate individual integrals if there is addition in the function you are integrating.

and whenever you add these integrals together, you only need to account for 1 of the constants (C).

So the integral of our velocity will go as follows:

or

We can solve for our constant, C, if we are given initial conditions, such as the golf cart was moving at y’(0) = 1 m/s when we began collecting our data.  Otherwise, we leave the integral in this form.  If you would like to learn how to perform a definite integral, refer to our article on definite integrals.

Vector dot product rules

Another simple review of the vector dot product, for those of you that have forgotten.  The operation that involves multiplying two vectors together can be done in a few ways.  The first operation is called either the scalar product or the dot product.  One of the well known definitions looks like this:

RULE 1:

This is a scalar product that is equal to the two magnitudes multiplied together and multiplied by the cosine of the angle between them.  If  the two vectors are perpendicular to each other then the angle between them is 90 degrees, which will make the dot product equal zero.  This is an equivalent equation.

RULE 2:

When you finish your dot product, you should have a number, not a directional vector.  So if you get something like this you did something wrong: .  If you ended up with (any #) then you don’t have to completely rule out your answer.

Practice vectors

A few vectors to practice with:

Okay, now we have found our dot product by applying RULE 2 above.  We can use this value along with our individual vector magnitudes to apply RULE 1 and obtain the angle .

We can calculate the projection of the vector A onto the vector B by this relationship:

Note that this is a scalar quantity, and that we can also define the projection of B onto vector A in a similar fashion:

Performing a vector dot product in MATLAB

Perform a dot product in MATLAB like so:

>>A = [ 1 2 3];

>>B = [2 3 4];

>>dot(A,B)

Enjoy

This is a very simple post and a very simple subject, but every once in a while even the experts need to be reminded how to do simple addition and subtraction with vectors.  Let’s go ahead and specify a couple vectors that we can work with.

Vector

In MATLAB: >>A = [1 2 4];

Vector

In MATLAB: >>B = [2 3 4];

When you add vectors together, you add each individual directional component ().  Subtraction works the exact same way.  Lets go ahead and do a few, C will represent the vector that results from the addition and subtraction.

These vectors are all in 3-dimensional space with a X, Y and Z component.  The number in front of each directional component is the weight or magnitude of that particular directional component of the vector.  There it is, short and sweet

This is how you calculate the volume of an ellipsoid with the following equation.

rmax = 1;
V = 0;
step = 0.02;
A = 5;

B = 10;
C = 15;
for x= -5 : step : 5
for y = -10 : step : 10
for z = -15 : step : 15
if((x/A).^2 + (y/B).^2 + (z/C).^2) < rmax
V = V + step.^3;
end
end
end
end
disp(V)

it should spit out , approximately

One of the most important calculus concepts to learn is derivation.  Here I will show you how to calculate some simple derivatives.  The derivative looks like this and can be read as “derivative with respect to x” so if you have a function and you are supposed to perform then you are going to be taking the derivative of the function “f” with respect to all variables “x”.

If n is a positive integer, this is your theorem:

let’s apply this to our equation .  Our derivative is looks like this to start where n = 1 because 5x is not raised to any power.

The first derivative will be represented like so:

or

Our equation is now equal to a constant.  If you take the derivative of ANY constant, the result is Zero.  So here

Let’s make this example a little more realistic, our new equation is

each piece of the equation loses 1 order ( second order), keeping in mind that the 3 was a constant and is now zero.

we can continue all the way to zero.

An example of finding the equation of a line

In practice you will often be asked to find the equation of a line that is passing through a given point and is parallel to another line.  A line is parallel to another line if they have the same slope (m).  I think the best way to show how to do this would be to do an example problem.

example:

Find the equation of the line L passing through the point (2, 3) and parallel to the line

The general equation for a line is

where (rise over the run), x and y are your variables, and b is the point where the line intersects the y-axis.

Because the line we are finding is parallel to the one given, we know they have the same slope, which is -3 here.  So far we know this about our new line:

the slope of the line is defined as the change in distance y divided by the change in distance x on a coordinate plane, it is written like so, .  So what can we do with this equation?

We know that our line is going to pass through the point (2 , 3), and where and are our known points, and .  This relation works because we are dividing the difference in distance between any point y and our known point, by the difference in distance between any point x and our known point, giving us an equation in terms of x and y.

if we simplify we get

We want this equation in the form y = mx + b so move the “-3″ over and our final equation is

Graphing the equation of the line

Let’s make sure we’re right.  The equation should have the same slope (m) as the equation we were given, which is -3 and this checks out.  We also know that we used our given point to and the slope formula to reach this conclusion, so all signs point to this being pretty darn accurate.  Go ahead and graph it to make sure.

Irrational Numbers

Any number that is not rational is called irrational.  Here are a few examples of some irrational numbers:

…

Every rational number can be written as a repeating decimal, such as is a rational number where as , above, is not repeating.  This is a quick and easy test with a calculator if you are confused as to whether or not a number is irrational.

Between any two real numbers there is a rational number and an irrational number.

Rational Numbers

In order to acquire a strong understand of the basic concepts associated with calculus, you should understand the real number system.  Real numbers fall into quite a few different categories including integers, rational numbers and irrational numbers.

A rational number is a real number that can be written as a quotient of two integers, where the integer ( 1 ,2, 3 etc) in the denominator is not zero:

r = where

also, ‘n’ is also a rational number in this case because

Examples of rational numbers

(a) (b) (c) (d)

example

is a rational number since

As you can see, rational numbers can be represented in an infinite amount of different ways, that is, you can keep incrementing the numerator and denominator appropriately  to acquire the same result.  Any number that is not rational is called irrational, click here to read about irrational numbers.

Linear Approximation

Here I will attempt to use linear approximation to estimate the value of

The first thing you need to do when performing a linear approximation is set up an equation with known numbers as variables:

here we have turned 1.07 into “x”.

Now take the derivative of either side; when x = 1 and dx = 0.07

ln(1.07) = f(1.07) because our f(x) function was equal to ln(x), so we know in this case, that x = 1.07, after all, it was given in the problem.