1. Introduction to Integration
Chapter 4. INTEGRALS
Introduction to Integration
Area and Definite Integral

2. 4.1 Area problem
We know how to compute areas of rectilinear objects, such as rectangles, triangles, polygons
How do we define and compute areas of more complicated regions (e.g. area enclosed by a circle)?
Idea: approximate such regions by rectilinear regions (for example, by polygons)

3. Area under the curve y=f(x) between a and b
Assume f(x) ≥0 on [a,b] and consider region R = { (x,y) | a ≤x ≤ b, 0 ≤ y ≤ f(x) } y
y = f(x)
f(x)
(x,y) x x a b
What is the area of R?

4. Approximation by rectangles
y = f(x) y x
x0= a x1 x2
xi-1 xi b
=xn
Divide [a,b] into n intervals of equal length
Use right endpoints to built rectangles (columns)

5. Area of i-th column is f(xi)•∆xy
f(xi)
y = f(x)
f(xi)∆x
f(xi) x a x1 x2
xi-1 ∆x xi b
=xn
x0=

6. Total area of all columns isy
f(xi)
y = f(x)
f(xi)∆x
f(x2)∆x
f(xn)∆x
f(x1)∆x x
x0= a x1 x2
xi-1 ∆x xi b
=xn

7. Definition. Area under the curve isy
y = f(x) x a b
n=14

8. Theorem. If f is continuous on [a,b] then the following limit exists:y
y = f(x) x a b

9. How to find xi ∆x ∆x ∆x x
x0= a x1 x2
xi-1 xi b
=xn

10. Using left endpoints
y = f(x) y x
x0= a x1 x2
xi-1 xi b
=xn

11. Area of i-th column is f(xi-1)•∆xy
y = f(x)
f(xi-1)
f(xi-1) x
x0= a x1 b x2
xi-1 ∆x xi
=xn

12. Total area of all columns isy
y = f(x)
f(xi-1) x
x0= a x1 x2
xi-1 ∆x xi b
=xn

13. Note: Ln ≠ Rn y x
x0= a x1 x2 xi b
=xn
xi-1

14. Ln - Rn = ?

15. Nevertheless…
Theorem. If f(x) is continuous on [a,b], then both limits and exist and

16. Using sample points
Choose a sample point - an arbitrary point xi* in [xi-1, xi] for each i
y = f(x) y x
x0= a
x*1 x1
x*2 x2
xi-1
x*i xi
x*n b
=xn

17. Area of i-th column is f(xi*)•∆xy
f(xi*)
y = f(x)
f(xi*)
f(xi*)∆x x
xi-1
x*i xi a x1 x2 b
=xn
x0= ∆x

18. Total area of all columns isy
y = f(x) x
x0= a
x*1 x1
x*2 x2
xi-1
x*i xi
x*n b
=xn

19. Theorem
If f(x) is continuous on [a,b], then the limit exists and does not depend on the choice of sample points

20. Definite Integral
Now we consider functions that may change sign on [a,b]
In this case, we need to take into account sign of f(x)
Idea: use “signed area”

21. Signed area y A1 A3
y = f(x) x a A2 b
“Net Area” = A1 – A2 + A3

22. Choose sample points x*i x*n x*1 x*2 y = f(x) y xi-1 xi x x0= a x1 x2b
=xn

23. Signed area of i-th column is f(xi*)•∆xy
y = f(x) ∆x
xi-1
x*i xi x
x0= a x1 x2 b
=xn
f(xi*)

24. Net area of all columns isy
y = f(x)
xi-1
x*i xi x
x0=
x*n a
x*1 x1
x*2 x2 b
=xn

25. Riemann Sum that correspond to n and given choice of sample pointsy
y = f(x)
xi-1
x*i xi x
x0=
x*n a
x*1 x1
x*2 x2 b
=xn

26. Definite Integral of function f from a to b is defined as the limit of Riemann sumsy
y = f(x)
xi-1
x*i xi x
x0=
x*n a
x*1 x1
x*2 x2 b
=xn

27. Theorem
If f(x) is continuous on [a,b], then the definite integral of function f from a to b exists and does not depend on the choice of sample points

28. Terminology
Upper limit
Integral sign
Integrand
Lower limit

29. Definite Integral in terms of area:y A1 A3
y = f(x) x a A2 b