1. ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department
Orhan TUĞ (PhDc)
Integratıon, fınıte sum and defınıte ıntegral 1 1

2. Upper and lower estimates of the area
Figure 5.1.9
Figure 5.1.8
Error analysis
Error analysis

3. Upper and lower estimates of the area in Example 2

4. Approximations of the area for n = 2, 4, 8 and 12 rectangles
Section 1 / Figure 1

5. Approximating rectangles when sample points are not endpoints

6. 5. 1 Sigma notation

7. Some important sum of sequences

8. The graph of a typical function y = ƒ(x) over [a, b]
The graph of a typical function y = ƒ(x) over [a, b]. (P)Partition [a, b] into n subintervals a < x1 < x2 <…xn < b. Select a number in each subinterval ck. Form the product f(ck)xk. Then take the sum of these products.

9. The curve of with rectangles from suitable partitions of [a, b]
The curve of with rectangles from suitable partitions of [a, b]. Suitable partitions create more rectangles with shorter bases.

10. Definite integration
***The definite integral of f(x) with respect to x

11. The definite integral of f(x) on [a, b]
If f(x) is non-negative, then the definite integral represents the area of the region under the curve and above the x-axis between the vertical lines x =a and x = b
If f(x) is continious on an interval [a,b], then its definite integral obver [a,b] exists.

12. Rules for definite integrals

13. 5.3 The Fundamental Theorem of Calculus
Where f(x) is continuous on [a,b] and differentiable on (a,b)
Find the derivative of the function:

14. The Fundamental Theorem of Calculus

15. Indefinite Integrals
Definite Integrals

16. Try These

17. Answers

18. Indefinite Integrals and Net Change
The integral of a rate of change is the net change.
but
If the function is non-negative, net area = area.
If the function has negative values, the integral must be split into separate parts determined by f(x) = 0.
Integrate one part where f(x) > 0 and the other where f(x) < 0.

19. Indefinite Integrals and Net Change
The integral of a rate of change is the net change.
If the function is non-negative, displacement = distance.
If the function has negative values, the integral
Must be split into separate parts determined by v = 0. Integrate one part where v>0 and the other where v<0.

20. 5.5 Review of Chain rule

21. If F is the antiderivative of f

22. Check by differentiation
Let u = inside function of more complicated factor.
Check by differentiation

23. Check by differentiation
Let u = inside function of more complicated factor.
Check by differentiation

24. Substitution with definite integrals
Using a change in limits

25. The average value of a function on [a, b]