Definite Integral

Objectives

To enable students to,

1. understand the process of anti-differentiation.;
2. recognize the problem of calculating areas bounded by non-linear function;
3. understand how the limit of the sum of rectangles may be used to calculate the area bounded by a function;
4. understand the meaning of $\textstyle \int \limits _{a}^{b}\displaystyle f(x)dx$;
5. calculate the area under a function between two extremes;
6. apply knowledge and skills relating to anti-differentiation to solve problems;
7. verify that the area bounded by the curve y=f(x), x=a, x=b and x-axis =$\textstyle \int \limits _{a}^{b}\displaystyle f(x)dx$

Prerequisites/Instructions, prior preparations, if any

Knowledge on plotting graphs, differentiation, mapping and computing functions.

Geogebra Resources

Download this geogebra file from this link.

1. File:Geometrical interpretation of definite integral.ggb
2. File:Property of definite integrals.ggb
   

Download this geogebra file from this link.

Evaluation

Evaluate following Definite integrals and give their geometrical interpretation:

1. ${\textstyle \int \limits _{2}^{5}(x+1)dx}$
2. ${\textstyle \int \limits _{2}^{3}xdx}$
3. ${\textstyle \int \limits _{1}^{4}(x^{2}-x)dx}$