What is the use of integration and differentiation?
Let us get into this with all interest. Yes integration is a reverse process of differentiation.
First let us put the question why do we need such a mathematical process called integration. Actually integration is nothing but the summing up of a lot, some million and million items.
Let me explain how.
Suppose you want to find the volume of a cone of radius r and height h.
Let the cone be seated such that its vertex get coincided with the origin and its height be along the x -axis.
In case of a cylinder the radius will be the same at all heights and so if you consider a small part both the sides of that part will have the same radius and so no problem in finding the volume.
But in the case of the cone, as we move away from the origin along x-axis the radius of the cone will be gradually increasing.
So as you consider a slice any where in the cone both the sides of the slice will not definitely have same radius. Is that ok?
Now calculus comes into play.
You choose a small slice in such a way that both the sides of the slice would have almost the same radius. It is possible only when you have a slice of negligible thickness. Such negligible thickness is denoted mathematically denoted as dx which means delta x tending to zero.
Once again note down the statement: delta x tending to zero. This means delta x is very so near to zero but not zero.
With such....... thickness, both sides of the slice would have the same radius.Let the slice of thickness dx be chosen at a distance x from the origin and let the radius of the slice be y at that position. Then the volume of our slice will be pi y^2 dx. Now imagine! Such slices, innumerable in number, can be got in the cone right moving from the origin and extending upto the total length (height) h of the cone. So we must collect all such slices and add their volumes to get the actual or total volume of the cone.
So we integrate the term pi y^2 dx within the limits of x ie 0 to h.
Now it becomes more essential to replace y interms of x. How can we do this? By using similar triangle concept, the ratio of the corresponding sides will be the same. So y/x = r/h
From this we can easily have y as (r/h) x.
Now replacing y we get the expression to be integrated (usually named as integrand) as pi (r/h)^2 x^2 dx. The formula for integral of x^n dx is given as x^(n+1) /(n+1).
So following this we get the integral value as pi(r/h)^2 (x^3 /3).
Next important step is supplying limit of x.
First upper limit h. This would give a value of pi(r/h)^2 (h^3 /3)
With lower limit 0, the value would become 0
Now the difference between these two values will be the required volume of the cone.
That comes to be 1/3 pi r^2 h. (cancelling h^2)
So interesting! See how much helpful the technique of integration in finding the volume of the cone!
By differentiation, we chop things into finer and by integration we collect all such finer.
Hope you have got a gist of the tremendous usage of the branch of mathematics, named as calculus.
First let us put the question why do we need such a mathematical process called integration. Actually integration is nothing but the summing up of a lot, some million and million items.
Let me explain how.
Suppose you want to find the volume of a cone of radius r and height h.
Let the cone be seated such that its vertex get coincided with the origin and its height be along the x -axis.
In case of a cylinder the radius will be the same at all heights and so if you consider a small part both the sides of that part will have the same radius and so no problem in finding the volume.
But in the case of the cone, as we move away from the origin along x-axis the radius of the cone will be gradually increasing.
So as you consider a slice any where in the cone both the sides of the slice will not definitely have same radius. Is that ok?
Now calculus comes into play.
You choose a small slice in such a way that both the sides of the slice would have almost the same radius. It is possible only when you have a slice of negligible thickness. Such negligible thickness is denoted mathematically denoted as dx which means delta x tending to zero.
Once again note down the statement: delta x tending to zero. This means delta x is very so near to zero but not zero.
With such....... thickness, both sides of the slice would have the same radius.Let the slice of thickness dx be chosen at a distance x from the origin and let the radius of the slice be y at that position. Then the volume of our slice will be pi y^2 dx. Now imagine! Such slices, innumerable in number, can be got in the cone right moving from the origin and extending upto the total length (height) h of the cone. So we must collect all such slices and add their volumes to get the actual or total volume of the cone.
So we integrate the term pi y^2 dx within the limits of x ie 0 to h.
Now it becomes more essential to replace y interms of x. How can we do this? By using similar triangle concept, the ratio of the corresponding sides will be the same. So y/x = r/h
From this we can easily have y as (r/h) x.
Now replacing y we get the expression to be integrated (usually named as integrand) as pi (r/h)^2 x^2 dx. The formula for integral of x^n dx is given as x^(n+1) /(n+1).
So following this we get the integral value as pi(r/h)^2 (x^3 /3).
Next important step is supplying limit of x.
First upper limit h. This would give a value of pi(r/h)^2 (h^3 /3)
With lower limit 0, the value would become 0
Now the difference between these two values will be the required volume of the cone.
That comes to be 1/3 pi r^2 h. (cancelling h^2)
So interesting! See how much helpful the technique of integration in finding the volume of the cone!
By differentiation, we chop things into finer and by integration we collect all such finer.
Hope you have got a gist of the tremendous usage of the branch of mathematics, named as calculus.
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