What Is Curve Tracing in Definite Integral

A definite integral is an entity that is acted upon as a function. This is the value for all the variations within the limits of a function. Hence, it is possible for us to represent it in the form of a graph. To do so, we need an understanding of the type of function that is being talked about. We need to find the pair of coordinates for which the function will stand true. And, then plot those points over a graph in order to get a curve. 

Curve Tracing

The curve tracing of a definite integral mainly depends on the function that is being integrated. Only by representing that function in the graph form can we trace the curve for the function within the certain given limits. Each type of function has its own set of graphs. And their representation also varies with the change in the value of variables taken. However, it is to be noted that the function to be found in the integral must be a continuous function. 

Identity Function

These are the functions that represent the same value for both the x-axis and the y-axis. The angle formed by this function with the x-axis and the y-axis is 45°. 

Constant Function

This is the type of function for which the value of either the x-axis or the y-axis always remains constant. It is represented as a constant and straight line parallel to either of the two axes. 

Comparatively, it is easier for us to find out the area that lies under these functions. These functions represent straight-limed geometric structures. Hence you can find the area under these curves using general mathematical operations. The case will stay the same even if you are to find the area within the given limits. 

Point of Inflection

The point of inflection is the point on a traced curve for which the property of concavity is for that particular point. It means that there is a change in the way that the curve is meant to be traced. If at a certain point, the function is f(x) > 0, then the curve will be traced concave upwards. In the other case, when the function is f(x) < 0, then the curve will be traced concave downward. 

This property for a given point for a certain function can be found by using the different values for a particular variable. The point at which there is a change in the sign for the outcome of that function will become the point of inflection for that particular function. 

The point of inflection can also be found if we are aware of the sign that is over the second derivative of a given function. This sign when associated with the first derivative of the given function will help us in getting to know that point of inflection. When the f'(x) is zero, the point of inflection is stationary, while in other cases, this point is not stationary. 

Tracing Curve

Polynomial Functions

These are types of representations that have a certain abnormal curve to it curved concave upwards or downward. This type of graph gives a perfect measure for the point of inflection for a function.

Modulus of a Function

The modulus of a function is the straight lines traced as a curve that is with an angle of 45° with the x-axis and the y-axis. For the negative entities, its mirror image for the negative quarter is taken into account.

Greatest Integer Function

The graph of this function represents the segments as the solution to the function. In this type of graph, the value at one axis of the plane is a multiple of another value on the other axis.

Signum Function

This is the type of function that will have different graphs based on the symbol that is possessed by the outcome of the function. 

Conclusion

Tracing the curve for a definite integral is similar to tracing a curve for a function. However, it is to be taken into account that the function in question must be a continuous function. If the function is not continuous, if it has a discontinuity in it, then it won’t be possible for us to draw a curve for it like it normally would. All that we need to do is find the coordinates between which we think that the curve might exist.