Definite Integral and Area Under the Curve

The integrals can be imposed on any mathematical function, be it increasing or decreasing, negative or positive. The value computed is used further in calculus in various applications. 

Every function in mathematics is based on variables. These variables have a constraint that is to be in a limited range of values. Thus, the boundary conditions are restricted for the function. Thus, in terms of integrals, the definite limits or the boundary used is fundamental, often referred to as the limit of integration. 

The mathematical function on which the integration is imposed is the integrand. Now, the variable with respect to integration being carried out is referred to as the variable of integration. Thus, the integral is the basic conceptual part to be noticed for any computation to be carried out. 

Thus, we can say that abf(x)dx is the definite integral for the given boundaries to be a and b, respectively. The mathematical function f(x) denotes the function on which integration is imposed and is integrand. The variable in this mathematical equation is x and thus, is the variable of integration. 

Definite Integral and Area Under the Curve

The definite integral, limited to a particular range of values or limits, can be represented on the coordinate axis. Thus, we can say that any function on which integration is being imposed, within a certain range of values, will uncertainly cover a definite area. Suppose there is a straight line given as a mathematical function parallel to the x-axis. Now the range is limited to a certain boundary. Thus, with the given boundary and the given function, we can see that a rectangle is formed wherein the function is one of the sides, and the range can be defined as one of the sides. Thus, this is the normal area of a rectangle formed with the help of a definite integral. Thus, the area can be computed for any mathematical function.

Now, depending on the number of functions, we can compute the area. The graph made will have a particular upper and lower function within a given range. Thus, compute the area using definite integrals by subtracting them from one and another. 

Hence, we can say that, Area under the curve=ab(Upper Function – Lower Function) dx.

Thus, the area can be computed using the above formula. Also, it may be possible that the functions can change their positions, that is, upper and lower functions, at a particular value. The area under the curve is defined and split at that limit to get the accurate area. 

Example

The mathematical functions were plotted on the coordinate axis system on the positive side. The graphs are observed within the range 0 to 2 such that the graph is plotted and only considered for the given range of the limits. Thus, the area under the curves is to be computed with the help of the given information. The given function is 3×2+2x+10; hence the area was computed for the given curve. 

Thus, from the definition, we know that the curve is restricted for the range 0 to 2. Thus, the area of the given curve is as, Area under the curve=02(3×2+2x+10) dx=[x3+x2+10x]02=(8+4+20)=28

Thus, the area under the curve within the given range is evaluated using the definite integral, and it is 28. 

Conclusion

The definite integral with the given limited range of the value can have an application wherein the area under the curve can be computed with the help of the given mathematical function. The area can be computed based on the geometry of the curves placed on the coordinate axis. Thus, the different graphs can be composed such that the area is driven out with the help of normal definite integration of the basic calculus.