Calculation of the area with the natural logarithm rule
Find the area of the region bounded by the graph of:
As a solution, the first thing to consider is the graph made with GeoGebra software:
To calculate the area we must consider the following integral:
To solve the definite integral it is necessary to solve the indefinite integral:
We solve the indefinite integral by the method of integration of substitution or change of variable, where we will take as U = x2+1:
We restate the integral in terms of the new variable (U):
If we look in the tables (formulas) of the basic rules of integration, the integral of du/u = Ln(u), so the integral would be:
Since the indefinite integral has been solved, it only remains to evaluate the result with the limits of integration from zero to three, applying the fundamental theorem of calculus as follows:
Conclusion and lessons learned
As we can see, being able to correctly apply the integration technique that best applies to calculate the area helps us not to have any complications at the time of solving the indefinite integral.
Bibliographic Reference
Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006