Aproximação geométrica de áreas sob curvas utilizando o GeoGebra: Uma proposta envolvendo funções para o Ensino Médio

Abstract

This work proposes an innovative approach to teaching the calculation of areas under curves in high school without directly resorting to integral theory. The proposal uses geometric approximation methods, such as rectangles and trapezoids, along with technological resources supported by the GeoGebra software to obtain approximations of the area below the curve y=f(x). The present proposal introduces the functions f(x)=x, f(x)=x², and f(x)=x³, defined over an interval [a,b], where the area calculation is performed through approximation by dividing the interval into n subintervals. This allows students to visualize and discuss the potential errors involved. The methodology combines practical manual calculation activities with GeoGebra, providing a comparative analysis of approximation methods. Students are encouraged to reflect on the accuracy of the approximations, recognizing through this approach that the greater the number of divisions into rectangles and trapezoids of the area to be calculated, the closer the estimated area will be to the true value as calculated by technological tools. This work not only develops mathematical reasoning but also promotes the use of technologies in teaching mathematics, making the learning process more dynamic and engaging. The conclusion highlights the importance of geometric approximations in area calculation, introducing students to the study of Riemann sums and integrals in advanced stages of their education.