Integral in physics






Definite integral

Definite integral is a way to find the area of a graph.

The only simple way to get the area as the product of width and length is for the rectangle.
If the shape is not constant, you can not get the area by simple multiplication. When a section of the graph is defined, we use definite integral to find the area of that section.

Integral

The mathematical definition of the integral is :

Try to divide the sections so that the width of the sections is infinitely small. When this sum is always close to a constant value, we define that value as the definite integral of a to b.
\[ (Area) =\, \int_{a}^{b} f(x) \; dx \]

How to definite integral

In order to integrate, you first need to know the derivative.
Do your own search for the differential method.

Integral can be said to be the inverse of the derivative. Suppose a function f(x) is the derivative of F(x) with ‘x’. In this case, integrating f(x) results in ‘F(x) + C(integral constant)’.
Whatever the integration process, if you can integrate the function, you can get the definite integral value as :

\[ \int_{a}^{b} f(x) \; dx \, = \, F(b) – F(a) \]

Examples of definite integrals often used in Physics

Speed-time graph Speed-time graph Area = Distance traveled for the duration of the interval

 

Acceleration-time graph Acceleration-time graph Area = The rate of change of speed over the duration of the interval

 

Force-time graph Force-time graph Area = The amount of received impact (momentum) during the time of the interval

 

Force-distance graph Force-distance graph Area = The amount of work done to move the distance of the section, the energy stored or released in the spring