Skip to content

Introduction to Reimann Integration

December 14, 2010

Introduction: Integration

Integration is a process in mathematics that can tell us:

  • The area of a curved 2-D object (the sides aren’t straight, and there is no simple formula)
  • The volume of a curved 3-D object (once again, the sides aren’t straight)
  • The velocity of an object if we know its acceleration at time t (which means the acceleration changes all the time, as does its velocity)
  • The displacement of an object if we know its velocity at time t (the velocity and displacement change over time, so there is no simple formula)
  • The pressure on an object deep under water (the pressure varies as we go down)

Reimann Integration: Intro

A real time example that help to understand Reimann Integration:

Assume that we are planning to travel in a car from a city to another and you are new to travel between those cities and are unknown of distance between them. But you are curious to know the distance. Only thing you have access to are a stopwatch and the speedometer of the car. Using this two things , shall we calculate distance between two cities?

Yes

Try this procedure :

  • Have a pen and paper ready, and start the stop watch and note down the speed shown in the speedmeter at every minute.
  • Continue this till you reach the destination city.
  • Simply multiply each of the speeds with the corresponding time intervals ( here 1 minute) and finally add them all. The an approximation to distance between the two cities is in front of you !.
  • Note that the time interval here is one minute, that  is you have noted down speeds at the end of every minute. This time interval may not also be uniform, that is, you may also note down speeds first at the end of half minute, then at the end of one minute and then at the end of 1 1/2, and so on.  In such a case, we should multiply speeds with the corresponding time interval. This also gives an approximation for the distance !.
  • If you want a better approximation, then try reducing the time interval, that is, note down speeds at every half minute or even every 1/4  th of a minute.
  • what we have done here essentially is integration.

Now we will formulate this mathematically. Let us assume that we started at time 0, and let f(t) denote speed at which the car travelled at time t. Assume that we stopped after 5 minutes to have a break.

Suppose at the end of each minute, the speeds are 70, 72,45,25 and finally at fifth minute there is a halt for break, so it is 0. Again graphically

In our procedure above we have multiplied speeds with the corresponding time intervals. So essentially, we are finding areas of rectangles with height equal to the speed and width equal to the time interval, and then finally we are adding the areas. So we are approximating area.

  • As already said we are trying to estimate distance, but here we are estimating area, so the area which we are trying to estimate is nothing but the distance travelled. So the actual area under the graph is nothing but the exact distance travelled in 5 minutes.
  • But from the knowledge of physics, we can say that differentiating distance with respect to time gives speed. So integrating speed gives distance. So by the above point and this one we can conclude that integration is nothing but finding area under the graph of the given function.
  • Here we are given speed as a funtion of time and hence on integrating with respect to time we get distance travelled.

By above example we can say that integration is nothing but area under the curve and this can be estimated using rectangles. This idea is the basis of Riemann integration theory.

Definition of Riemann Sums and Integral:

Example:

Consider a function y=1 − x2

. Find the area between x=0.5 x=1?

This is the area we are trying to find:

Since n = 5, the width of each rectangle will be:

We aim to find the sum of the areas of the following 5 rectangles:

Now the height of each rectangle is given by the function value for that particular x-value.

For example, since y = f(x) = 1 − x2, the first rectangle has height given by:

f(0.5) = 1 − (0.5)2 = 0.75

It has area given by:

Area1 = 0.75 × 0.1 = 0.075

The second rectangle has height:

f(0.6) = 1 − (0.6)2 = 0.64

The 5th rectangle has height

f(0.9) = 1 − (0.9)2 = 0.19

Adding the areas together gives us the following. (We are writing it using summation notation, which just means the sum of the 5 rectangles. Also, we are adding the heights first then multiplying by the width, which is the same for each rectangle.)

In the above answer, we are finding the area of the “outer” rectangles. To find a better approximation, we could also find the area of the inner rectangles, add them and then average our result. The graph for the inner rectangles is as follows:

And this is the sum of the areas for the inner rectangles (the 5th one has height 0, so area 0):

The average of the 2 areas is given by: (0.245 + 0.17)/2 = 0.2075.

4 Comments leave one →
  1. December 18, 2010 6:14 am

    Dear Dhasthagheer,

    Really wonderful explanation.

    But I feel, you didnt conclude it in proper manner.

    Even it should be just two lines, conclude this post.

    then it should be awesome. 🙂
    😛

    My wishes,
    Arulalan.T

  2. December 19, 2010 8:37 am

    Reimann Dhastha 😉

    Kanukku puli aitingalo… 😛

  3. H.T.Phuc permalink
    June 2, 2011 3:29 am

    Dear Dhastha ! The topic is wonderful , I will support to you.
    Follow normal integration, my result is ~= 0,208333333…. ~= 0.2075 of Reimann integration.

Trackbacks

  1. Reimann Integration visualization using Mayavi2 « Li(G)NUx

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: