Areas in Definite Integral

Without the use of integration, it is not possible for us to find out the area under the curve. To perfectly calculate the area that lies under a curve with the greatest accuracy, that is why we need integrals. Using definite integrals, we find the area of a curve in between two given limits. To find the definite integral, there are certain conditions that need to be imposed on this. Like, the function must be continuous between the given limits and such.

Continuous function

A continuous function is a function that does not have any significant or outrageous change in its value. We can check the continuity of a function if we can draw out different values for that function only by making small changes in the values that are inserted in the function. 

The continuity of a function can also be checked by the medium of limits. The expression to check the continuity of a function can be given as follows:

Lim_x→c f(x) = f(c)

It can also be said that if the left-hand limit, right-hand limit and the value for x = c for a function exist and are the same, then the function can be said to be continuous at x = c.

A function is continuous at a point if the value for it can be defined at that particular point, and the limit at that point is equal to the limit of the given function. 

However, if it is found out that the function is not continuous at point ‘c’, then it won’t be wrong to say that the function is discontinuous at ‘c’. In this case, the point ‘c’ can also be referred to as the point of discontinuity. 

Non-negative

Non-negative functions are the functions to which the outcoming value will always be equal to or greater than zero. The values used for the function can be positive, negative or zero. However, It won’t have any effect on the symbol of the outcome. The result will always continue to be a positive entity. 

The nonnegative function plays a really crucial part in the concept of integration. Most of the functions that are to be found in the definite integral are non-negative functions. However, the perspective can change about these considering the value of the is above the curve of the given function, or the outcome is below that curve.

Let us assume there to be a function for which its limit does not exist at [ 0, ∞ ]. Now, this function is to be integrated. Its integration will not give a specified value if the function continues to be negative. Hence let us assume that the function is nonnegative. We are likely to be able to find out the derivatives of the given function. If the first derivatives of that function are in approximation to maxima of the given function, then it can be said that the limit exists for that function. In this case, it will be possible for us to integrate the given function. 

Definite integral area

Definite integrals are used to find the area that lies above, below or between two curves. This integral gives us the area that exists between a curve and the x-axis for a certain set of limits. It covers the extent of the area that lies in between those values. 

The fundamental theorem of calculus is applicable to find the area of a function using definite integrals. To utilise this method the difference between the antiderivatives of the given function when the upper limit is placed in those and when the lower limit is placed in those. The outcome of this action is the definite integral of the given function between the given limits. 

When we measure the area between a curve and the x-axis, the outcome of this operation is always a positive entity. However, in the case when the area to ve found lies above the curve and is in no jurisdiction under the x-axis, then it will be negative.

Conclusion

Using the fundamental theorem of calculus, it becomes much easier for us to calculate the area of a curve. To initiate this action, all we need is the limits for the function to which that curve belongs. By using those limits, we can find the area that lies between those limits. However, the given function should be continuous in between the limits. If that is not so, it won’t be possible to integrate that function. Also, the outcome remains the same if the function is negative.