Skip to main content

WHY CALCULUS -II

 Consider the y=x^2 curve


ADVT:  REFURBISHED MOBILES IN AMAZON

    What is the area under the curve? If it is a rectangle or circle, we can easily calculate the area. But it is an irregular shape.

    Let us do it in the following way.



Let us draw small rectangles under the curves and fill the entire curve with rectangles. Let us now calculate the area of each rectangle and total all the areas.  That will give the area under the curve. The smaller the width of the rectangles, the more accurate the result will be. We can do this process easier by writing a small computer program. The width of the rectangle is taken as 0.01 or even 0.000001. We can get accurate results.

    What they have done before the arrival of computers. They invented calculus. The 'integration' operation in calculus gives an accurate answer.

    The equation of the curve is y=x^2. The integration of x^2 is  2x^3/3. Suppose, you want to find the area between x=0 to x=3. Apply these limits to the formula.

Applying lower limit, 2x^3/3 = 2*0^3/3 = 0

Apply upper limit , 2x^3/3=2*3^3/3= 2*27/3 =18 unit.

The area under the curve is 18-0=18 units.

    Hence they have devised so many beautiful formulas in calculus. That makes physics and engineering easy.

    The 'Differentiation' in calculus can also be explained in this way.

    The small python program for integration is given below.

  1. import math
  2. lower_limit =0
  3. upper_limit =3
  4. witdth=0 .00001
  5. total_area=0
  6. x=lower_limit
  7. while x is less than upper_limit:
  8.       x=x+width
  9.       y=x**2
  10.       area=width*y
  11.       total-area=total-area+area
  12. Print(total_area)

   You can give the equation of any curve in the high-lighted area and verify the result. We have to appreciate the math in this way without fearing it.

Comments

Post a Comment

Popular posts from this blog

THE EARTH, A SUPER ORGANISM

     JOIN MY COURSE: "Become a programmer in a day with python"       A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess.      It says, "Earth is a self-regulating organism like a human being.  The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment".  Hence the earth is still suitable for the life to thrive.      Imagine, in a particular place, there are lot of flowers.  Some flowers are white and some are darkly coloured.  We know, white reflects light and heat while dark absorbs the same.  White flowers can thrive in hot climate.  But dark flowers requires cold climate.  The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist.  This is the essence of "Gaia" theory.      On our earth, ...
3 comments
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.   ...
Post a Comment
Read more

DISORDER IS THE "ORDER OF THE DAY"

         Imagine a balloon full of air.  The air molecules are moving randomly inside the balloon.  Let us pierce the balloon with a pin.  The air rushes out.  Why should not the air molecules stay inside the balloon safely and ignore the little hole?  That is not the way the world works.  The molecules always "want to occupy as many states as possible".  Hence the air goes out in the open to occupy more volume.   The things always goes into disorder (entropy) and the disorder increases with time.  The above statement is what we call "second law of thermodynamics".      Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze.  Freezing means bringing things to order and arrangement.  It violates the second law.  Hence it will never happen.  Hence heat must flow from high ...
Post a Comment
Read more