Concept of Integration

Concept of Integration

Integration is a way of joining small pieces together to get the total amount.

Integration is a method of adding or summing up the parts to find the whole.

It is like building a whole house by putting all the blocks together.

This process is the direct opposite of differentiation because it helps us find the original formula when we only know how it changes.

It is a reverse process of differentiation, that is, it is used for finding functions when the derivative is given.

In simple terms, if differentiation is 'breaking down', integration is 'fixing back'.

We use integration to calculate the space inside shapes, the size of containers, and many other important things.

Integration can be used to find areas, volumes, central points and many useful things.

It helps us measure things that do not have straight sides.

Example of the Reverse Process

If y = 3x + 9, dy/dx = 3

If y = 3x + 2, dy/dx = 3

If y = 3x, dy/dx = 3

Why we add a constant

When we differentiate different formulas, they can give us the same answer because the numbers at the end become zero.

This means that the integral, 3 can be 3x + 9, 3x + 2, 3x, etc. Because of this, when you integrate, you have to add a constant. So the integral of 3 becomes 3x + c, where c is a constant.

We use the letter 'c' to stand for any number that might have been there before.

Integration Symbols

We use a special long curly sign and a small 'dx' to show we are doing integration.

An "S" shaped symbol (∫) is used to mean the "integral of" and dx is written at the end of the terms to be integrated, meaning "with respect to x". This is the same "dx" that appears in dy/dx

The curly sign tells you to start, and the dx tells you which letter you are working with.

Why we study integration

The concept of integration is developed to:

find the problem function when its derivatives are given.

find the area bounded by the graph of a function under certain constraints.

Standard Integrals