Imagine a growing circle. What is the 'most-least' area you can add to the circle. or what is the area of the thinnest ring that can be added to the circle. It is 2*pi*r. How? Consider a rectangle. It's area is length*breadth. Suppose breadth vanishes, the area is only length. Similarly length of the thinnest ring is 2*pi*r-- the circumference. Also we can say 2pi*r is the infinitesimal change in the area of the circle.
From Dot, if you go on add big and bigger rings, a circle will be formed. If we add the areas of the all the rings, we will get area of the circle. In calculus adding is called "integrating". Hence, integrating '2pi*r' from 0 to r we get,
We got, total area by integrating small change in area. Calculus works.
Add circular sheets of diminishing area one above the other, you get a solid cone. correct? Refer figure. Here, the least decrease in volume (by the same argument) is pi *r^2. By integrating pi*r^2, we get 1/3pi*r^3. That is the volume of the solid cone. Again calculus in action.
The reverse of integration is differentiation:
By differentiating volume, we get area. By differentiating area, we get length- circumference.
Differentiation:
We know Y = x^2 gives parabola. By differentiating, we get 2x- which is the small change in Y at any point x. 2x also gives the slope at any point in the curve.
Y=x^2
dy/dx = 2x
Hence, differentiation of Y with respect to x is
1. A infinitesimal change in Y when change in x reaches the limit zero.
2. A change in Y for unit change of x, for any value of x.
3. Rate of change of Y with respect to x.
4. A gradient or slope at any point in the Y-X curve.
Integration:
1. Integration is the summation of small changes.
2. Commutative effect of gradual or continuous small changes.(increments or decrements).
3. It gives area under the curve between two limits.
4. It also gives volume enclosed by curvy sheets.
WHY CALCULUS
1. For discrete and countable things, Asthmatics and Algebra is enough.
2. When things have mathematical relationships and when things changes continuously, calculus is required.
For counting money, calculus is not required. But for finding inflation rate, calculus may be required.
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From Dot, if you go on add big and bigger rings, a circle will be formed. If we add the areas of the all the rings, we will get area of the circle. In calculus adding is called "integrating". Hence, integrating '2pi*r' from 0 to r we get,
We got, total area by integrating small change in area. Calculus works.
Add circular sheets of diminishing area one above the other, you get a solid cone. correct? Refer figure. Here, the least decrease in volume (by the same argument) is pi *r^2. By integrating pi*r^2, we get 1/3pi*r^3. That is the volume of the solid cone. Again calculus in action.
The reverse of integration is differentiation:
By differentiating volume, we get area. By differentiating area, we get length- circumference.
Differentiation:
We know Y = x^2 gives parabola. By differentiating, we get 2x- which is the small change in Y at any point x. 2x also gives the slope at any point in the curve.
Y=x^2
dy/dx = 2x
Hence, differentiation of Y with respect to x is
1. A infinitesimal change in Y when change in x reaches the limit zero.
2. A change in Y for unit change of x, for any value of x.
3. Rate of change of Y with respect to x.
4. A gradient or slope at any point in the Y-X curve.
Integration:
1. Integration is the summation of small changes.
2. Commutative effect of gradual or continuous small changes.(increments or decrements).
3. It gives area under the curve between two limits.
4. It also gives volume enclosed by curvy sheets.
WHY CALCULUS
1. For discrete and countable things, Asthmatics and Algebra is enough.
2. When things have mathematical relationships and when things changes continuously, calculus is required.
For counting money, calculus is not required. But for finding inflation rate, calculus may be required.
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