Calculus 2.0 _ Integration

You lasted the roller-coaster once... Enjoyed the thrill, but then you realise it's one of those rollercoaster which goes back in reverse and now you are debating if your stomach will support your ride or your heart will explode in the happy adrenaline rush, Welcome to the world of Calculus! (haven't read the first part yet? Click this link!

https://mohanapriyawrites.wixsite.com/studiedyes/post/calculus__differentiation

)

Multiplication has Division, Subtraction and Addition , leaving Differentiation alone is just cruel isn't it?

Just like how division is the opposite of multiplication and subtraction the opposite of addition, Intergration is the reverse process of differentiation.

And by opposite, I mean opposite in everything...

So for differentiation, we saw that it helped in finding the slope of the graph, Integration will help us find the area under the graph

We will be finding the area of the blue part using the concept of Integration

Ofcourse, We know how to find the area of a rectangle, area of a triangle ( and for the little adventurous people reading this blog maybe even a trapezium) , but area of a curve? How?

Let's express the area under the curve into components whose formulas our 4th grade maths teacher worked hard for us to understand

So break the curve into small rectangles, each with very, very very small breadth (Almost negligible.. tending towards 0 kind -- consider this dx )

Consider 'y' to be the length (as it is a variable, its value keeps changing for every small rectangle, we are generalising it as 'y' here)

Area of the blue part will be = Area of Rectangle = length * breadth

Hence, Blue part = y*dx

The yellow part will have a similar area ( as it is very close to the blue part)

So when we keep breaking the curve into miniscule rectangles and add the areas of these thousand rectangles, we will get the area of the curve

No, don't panic, I am not going to ask you to add a thousand values one by one, This is where integration kicks in.

If we represent the area of each miniscule triangle as dA, we get

(The fancy S, or snake or (however you percieve it_ text in the comment box what it makes you think of! _ is the symbol of integration)

Now, that the introduction part is done, Let's get down to buisness

Formulas related to Intergration

As I have mentioned before, Integration is the opposite of the Differentiation, so we will do the exact opposite of what we did in Differentiation in integration

(compare this fomula with the one for diffentiation, you will see that in differentiation, the power is decreased by one, then multiplied, Here we increase the power by one ,then divide)

But unlike differentiation, there are two types of Integration

1) Definite Integration

2) Indefinite Integration

These are based on whether the limits to the integration are given or not (Limits are basically which tell us from which value we should start and with which value we should stop the integration__ In the graph given in the start, there was a definite point from where we started, and a definite end point-- this is a definite integration, now if we extend the graph till infinite on both sides and try to find the area -- that is infinite integration)

Definite Integration:-

In a definite Integration, the upper limit ( 2, in the given example) and lower limit ( 1, in the given example) are specified. To calculate the integeration, we follow the formla first and integrate the variable, then we substitute the upper limit and lower limit in the integrated form. Later, we should subtract the upper limit's value from lower limit's

(An example will help you understand these crazy steps clearly, dw)

Indefinite Integration:-

Indefinite integration is comparatively easier, We need to just integrate it then add the constant C to the end answer. C is called constant of integration (yup, not too obvious is it?) . As the limits are not known, the Constant of Integration should be added to make the solution valid.

Saying Integration is a vast topic would be an understatement, this blog covers the basics of integration... but we have just scratched the surface. There are so many uses of integration in our real life and its majorly used in proving other theories and formulas used almost everyday (We know formula of triangle is 1/2 * height * base, but ever wondered why?... Integration will help us prove that ( now, that's for another blog)