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WHY CALCULUS -II

 Consider the y=x^2 curve


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    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.

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