1. Use integration to find the value of

2. ► The area of a trapezium is given by the formula a b h

3. Inspirational Education

4. Numerical Integration Objectives: Some - Explain how to improve your estimate or compare it with the true answer Some - Explain how to improve your estimate or compare it with the true answer Most - Confidently work with a calculator to obtain a correct answer to the estimate Most - Confidently work with a calculator to obtain a correct answer to the estimate All - Use the trapezium rule method to find an estimate for an area under a curve All - Use the trapezium rule method to find an estimate for an area under a curve

5. The simplified Bilborough model

6. Double Trouble Total area =

7. The story so far.... ► We found an estimate for the area under the curve using 6 trapeziums (83.5) ► We then found an estimate for the area under the curve using 12 trapeziums (83.88) ► The exact area by integration is...

8. Language used in exam questions

9. Numerical methods ► Sometimes a function can be too tricky or impossible to integrate. In cases like this we use a numerical method to get an approximate solution. ► One of the most common methods of doing this is using the Trapezium Rule.

10. Trapezium Rule If we wanted to find the approximate area under the following curve between 0.5 and 2.5, we could split it into trapeziums. We have split this area into four trapeziums, so we would say we are using 4 strips of width 0.5

11. Generalising.. y0y0 y1y1 y2y2 y3y3 y4y4 hhhh Area = ½ h (end ordinates + twice sum of ‘interior’ ordinates)

12. The big formula ► n is the number of strips used ► h is the strip width and the formula on the extreme right tells us how to find it Area = ½ h (end ordinates + twice sum of ‘interior’ ordinates)

13. Washing Line ► Find as many answers as you can - you will be more likely to learn the technique ► Arrange the items on the washing line in ascending order from left to right - this will help your checking skills ► Talk to other people and share your ideas - listen too!

14. Example Using the Trapezium Rule with 5 strips, calculate an approximation to:x0.511.522.53y Is the answer an overestimate or an underestimate??

15. 5.146 0.8548 22.39 11.2 12.77 2.034

16. Accuracy If the number of strips is increased, the accuracy of the estimate improves.

17. Accuracy Whether the estimate is too large or too small depends on how the gradient is changing. Here, the estimate is an overestimate of the true area under the curve Now, the estimate is an underestimate of the true area under the curve