How to Find the Area Under Curves

The integral calculus is a branch that helps us to compute the values for different functions and relations. One of the applications, which is widely used in different sectors and domains is the computation of area covered by any curve when plotted on the coordinate system. The area under the curves refers to the area that falls between two reference boundary points extended as the axis. 

Using the integrals, the value of the area under the curves can be computed easily. For any given mathematical function, say f(x), it is necessary to be aware of the range or the two reference points and the variable of integration for the curve. With the help of these parameters, one can easily compute the area under any given curve. 

Geometrically a definite integral represents the net algebraic area of a region under a curve or graph of the function. Now that we have understood the meaning of the area under the curves, let us look at some important methods to find the area under the curve as that can enhance our understanding.

Basic methods to calculate the area under curves

First method:  

The first method is boundary rectangles. The area under the curve that is to be evaluated is broken down into small rectangles. The smaller the rectangles, the greater is the accuracy of the area calculated. The summation of all these rectangles gives us the area under the curve. This method is highly effective for curves that are linear and regular in nature. As the curve gets irregular, the rectangles drawn can be visualised to not fully cover the area of the curve, thereby missing some areas. Here all areas obtained can be considered as positives and added up to find the net area under the curve. 

Second method of the definite integral calculus: 

The area under the curve can be determined using a definite integral if the boundary points (starting and ending points of the curve) and the equation of the curve are known. 

Hence, the change in the value with respect to the variable is made as small as possible which marks the introduction for the ‘dx’ and ‘dy’. Thus, conceptually, this small change in value is represented and is added to the integrals. The addition will be made from the smallest rectangle (drawn to calculate the area under the curve) possible.

This will lead to the near elimination of the area that was missed or not calculated while computing using bigger rectangles. And this will give us the entire area under the curve. However, it must be understood that the area obtained is the algebraic area and not the net area. For definite integral, subtract the area under the negative axis of the variable. Therefore, to obtain the net area, boundary points must be broken where the curve is below the variable axis and modulus value should be added. 

Concept of arbitrary constant in calculating the area under curves

When it was first learned about how to find indefinite integrals or better termed as antiderivatives, adding an arbitrary constant was necessary. This was an essential step to represent the family of curves that can be obtained by varying the value of an arbitrary constant.

However, there is no need for an arbitrary constant when solving a definite integral, since it represents the area under a fixed curve in a fixed interval. Now once a definite integral is interpreted as an algebraic area, it becomes quite clear that its value is particularly a fixed number which is the number of units that are in the area of that particular region.

Formula for calculating the area under curves

The area under simple curves, the area bordered by a curve and a line, the area between two curves, and the volume of solids may all be calculated using definite integrals. Integrals are also used to solve problems involving displacement and motion.

                     Area under the curve =ab(f(v)−g(v))dv calculated over a to b

Where f(v) and g(v) are the curves between which we have to calculate the area.

Application

The concept of the area under the curve finds usage in our everyday life. 

• A variety of dam and bridge constructions require an in-depth soil analysis for which the area under the curve needs to be determined. 
• It also helps determine the area of irregular objects that are stitched together in a two-dimensional plane. 

Conclusion

Integral Calculus provides us with an easy way of calculating the area under the curve. The area under the curve has various important applications to our real world. The trajectory of space satellites is determined by calculating the area swept by curvilinear imaginary orbits.  It also helps us optimise the area usage which is especially useful for architects and engineers. The area under the curve also represents our substantial knowledge about the two-dimensional world and its existence. Such an area can be easily derived using integration.