Question

What is integral and differential calculus?

How it can be used in numericals?

Please explain with an example each.

Is it very necessary to learn more detailed about them keeping in mind the AIIMS UG examination?

Answer 1

Imagine pushing a car up a hill. If the slope is always the same, then you’ll always have to push just as hard, but if it starts out steep and then starts to level off then the effort you have to put in will be constantly changing. So how can you work out how much energy it will take for you to push the car to the top of the hill?
One way to make it easier is to divide the problem up. If you divide the curve of the hill up into lots of small sections, each one of those small sections is almost straight, making the problem much easier.
Calculus is essentially a way to split up problems like this even further. In fact, it splits them up into infinitely small sections, giving you an exact answer.

There are two main tools used in calculus: differentiation and integration. To understand what these are, it helps to understand functions. A function takes an input and applies a rule to it to give an output. For example, it might take the input x and square it; this would be written as f(x)=x2f(x)=x2
Differentiation is used to find how much something changes. For example, if you have a function which tells you how fast a car is going a certain time after it sets off, differentiation can tell you its acceleration. The result of differentiation is known as a derivative.
Integration is the inverse of differentiation. Given the function described above, it could be used to find the distance the car travelled over a certain period of time

Integration works out the area underneath the graph between two limits, as shown by the shaded area between 0.5 and 1. What this area represents depends on what the axes of the graph mean: for example, if the x-axis was time and the y-axis was speed, the shaded area would be the distance covered in that period of time. Again, you can estimate this by dividing the graph into sections whose areas you can work out using simple formulas, but integration can give you an exact answer.