Introduction to Integration Chapter 5. INTEGRALS Introduction to Integration Area and Definite Integral

5.1 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)

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?

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)

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

Total area of all columns is y 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

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

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

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

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

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

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

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

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

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

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

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

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

5.2 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”

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

Choose sample points x*i x*n x*1 x*2 y = f(x) y xi-1 xi x x0= a x1 x2 b =xn

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

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

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

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

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

Terminology Upper limit Integral sign Integrand Lower limit

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