1. MTH1170 Integrals as Area

2. Integrals as Area
Using the findings from The Fundamental Theorem of Calculus we can calculate the area under a function between the variable axis, and between some specific bounds.

3. Integrals as Area
But how is the area actually calculated?

4. The Riemann Sum
In mathematics, a Riemann sum is a method used to calculate the area under a function. One very common application is to approximate the area of functions or lines on a graph, but also the length of curves.
To calculate the area under a function using this method we divide the area under f(x) and between a and b into “n” equal sub sections and create a rectangle in each one.

5. The Riemann Sum
Then in order to approximate the area we can calculate the area of each rectangle individually and add them all up. Mathematically this is done with the sigma symbol.

6. Riemann Sum

7. Riemann Integral
If we take the limit as the number of rectangles goes to infinity, the Riemann Sum becomes the Riemann Integral:

8. Area of Plane Regions
So far we have found that the area under a function can be computed using a definite integral. But we haven't yet learned some of the limitations associated with the definite integral.
We know that the definite integral calculates area above the variable axis, but what about area underneath the variable axis?

9. Area of Plane Regions
Whenever there is area between the function and the variable axis that falls underneath of the axis, this area is treated as negative by the definite integral

10. Area of Plane Regions

11. Example
Find the area between the function and axis from x=a to x=b.

12. Example
Here we can integrate from a to c treating the area as negative, and integrate from c to b treating the area as positive:

13. Example
Find the area between the x-axis and the function y = x^2 - 9 from -5 to 5.

14. Example

15. Example
Find the area between the x-axis and the function y = cos(x) from 0 to 3pi/2.
cos(x) has zeros at x = pi/2 and x = 3pi/2.

16. Example