Integration

1. Area under Curve Integration

2. History • Calculus was historically developed to find a general method for determining the area of geometrical figures. • When these figures are bounded by curves, their areas cannot be determined by elementary geometry. • Integration can be applied to find such areas accurately.

3. Trapezoidal Rule • Also known as Trapeziod/Trapezium Rule • An approximating technique for calculating area under a curve • Works by approximating the area as a trapezium

4. Trapezoidal Rule (2, 4) (1, 1) From diagram, clearly, it is an overestimate. Actual Area = 2.67 units2.

5. Using Integration to find exact area

6. Using Integration to find exact area

7. Why?

8. Using rectangles to approximate the area under the line - Download Geogebra File

9. Using rectangles to approximate the area under the line Dividing the area under the line into 4 strips, We will start to approximate the area by finding the area of the rectangles Width of each rectangle = 0.25

10. What about n strips? width of each rectangle = Find the height of each rectangle Write down the statement for the area of each rectangle and sum them up 0

11. What about n strips? Dividing the area under the line into n strips, width of each rectangle = 0

12. What about n strips? As we increase the no. of rectangles, the white triangles will be filled up by the rectangles and we will get a better approximation of the area. 0

13. What about a curve? Similarly, we divide the area under the curve into n strips. width of each rectangle = Find the height of each rectangle Write down the statement for the area of each rectangle and sum them up

14. What about a curve? Similarly, we divide the area under the curve into n strips. width of each rectangle =

15. What about a curve?

16. Area by Integration

17. Find the area under the curve Example between x = 3 and x = 6 Find the area under the curve between x = 3 and x = 6