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

2. . A
Figure 5.1.9
Figure 5.1.8

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. The graph of a typical function y = ƒ(x) over [a, b]
The graph of a typical function y = ƒ(x) over [a, b]. 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.
5.2

8. The curve of with rectangles from finer partitions of [a, b]
The curve of with rectangles from finer partitions of [a, b]. Finer partitions create more rectangles with shorter bases.

9. 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

10. Rules for definite integrals

11. 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:

12. The Fundamental Theorem of Calculus

13. Indefinite Integrals
Definite Integrals

14. Try These

15. Answers

16. 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.

17. 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.

18. 5.5 Review of Chain rule

19. If F is the antiderivative of f

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

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

22. Substitution with definite integrals
Using a change in limits

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