Importance of area under the curve

Antiderivative or integration is the addition of seemingly infinitely small data put together by the process of integration. Tracing all the way back to Greek integration has been a part of the maths curriculum long back; however, it was during Newton’s time that it was realised that both integration and differentiation are inverse operations. So the complex shapes with no definite form are calculated using the form of integration; for the easy shapes, the formula is always valid. However, the indefinable structures are calculated with the help of integration.

The integration sign is represented by “∫”.

Calculation of the area under a graph

Antiderivative or integration of a graph helps to calculate the area of shapes that cannot be defined or those that are not bound by certain. It is certainly trickier to calculate when curved surfaces are involved. So basically, if we need to calculate the area of the curve given below:-

Method of finding the area under the curve

We need to solve this by the following methods if the curve is given according to the x-axis, such as  y=f(x). So, the formula of finding the area under the curve y is shown below:

abf(x)dx

The area calculation takes place in the following steps:

A=baydx

A=baf(x)dx

A=[g(x)]ba

A=g(a)-g(b)

Similarly, for the y axis the converse is; however, the function is x=f(y):

A=baxdy

A=baf(y)dy

A=[g(y)]ba

A=g(a)-g(b)

Here g represents the integrated function of the equation. However, for the calculation of this, we need to have some data or values already provided with the graph, the equation, and the values like a and b. 

Application in physics

As we all know, physics is applied maths; the use of integral calculus and the area calculation is very important for calculating the values of some physics values, especially in the Newtonian laws or subjects derived from that like kinematics and work. Both of which are amazing examples of the uses of the integral calculus of area under the curve.

So the main applications on physics are

1. A line integral is used whenever it is required to find out how much of a vector has accumulated along a path. The classic example is that work is expressed as the line integral W=Fdx

W=Fdx captures the idea of “accumulating work along the path”, but the concept can show up in a lot of places (the arrows determine the vectorial factors and the calculation and are shown so that they are not to be confused with scalar products).

2. Gauss’s law is not phrased in terms of a line integral but rather a surface integral. Here, the concept is capturing “I have a ball, and I need to calculate how much electric field is leaving the ball”. This is done by calculating the little amount of electric field leaving each section of the ball and then adding it all up. This is naturally done with a surface integral.

3. Generally, if one has a group of varying vector quantities, and it is required to calculate numbers with them over a distance that is larger than a single point, then it is likely to do some sort of vector calculus. The applications of area integral in physics are discussed in detail below.

1. Work Done by a variable force:-

W=Fx is a specific case of the basic work relationship that only applies to continuous force along a straight line. The area of the rectangle displayed, where the force F is plotted as a function of distance, is determined by this connection. The work can still be estimated as the area under the curve in the more general example of a force that changes with distance. The area under the curve, for example, can easily be computed as the area of the triangle for the work done to extend a spring. Since the integral of the force across the distance range equals the area under the force curve, calculus could also be used:

Work = 0xmF(x)dx = 0xmkx dx =12kxm2                                                             

2. Gauss’s Law 

Even though Gauss’s law is not directly about the electric field, it is about the electric flux; it is tremendously useful in determining expressions for the electric field. We can determine the electric field from the knowledge of the electric flux in instances when the charge distribution has particular symmetries (spherical, cylindrical, or planar). We can identify a Gaussian surface S, En=E in these systems across which the electric field has a constant amplitude of En=-E is also true if E is parallel to n everywhere on the surface. (On the surface, if E and n are antiparallel everywhere then value becomes: 

=sE.ndA=EsdA = EA = qenc0

Here the letters stand for, 

E= electric field, 

n= number of turns in the coil, 

A= Area of the surface, 

q= charge enclosed

             0= permittivity of the vacuum.

3. Vector calculus

In vector calculus, in which there are both calculations related to space and the necessary positions of a body in the 3-dimensional space and its velocity and other vectorial programs related to space and dimensions are a part of the calculation of both the curriculum of maths and physics.

 Conclusion

The application of integrations is determined by the sectors in which this calculus is utilised. In integration under a curve is generally used to find the area of shapes that are not properly defined even if so are so under a periodic motion having an amplitude or under the condition of some sine or cos functions. Such functions are difficult to calculate without the aid of integration so we need to calculate using integration rather than depending on the old formulas or methods.