Class Notes (1,100,000)

US (460,000)

UC-Irvine (10,000)

MATH (1,000)

MATH 2B (600)

ERJAEE, G. (20)

Lecture 3

Department

MathematicsCourse Code

MATH 2BProfessor

ERJAEE, G.Lecture

3This

**preview**shows half of the first page. to view the full**3 pages of the document.** MATH 2B - Lecture 3 - Definite Integrals

Definition. Suppose that f is a function defined on an interval [a,b]. Let n be a positive

integer, define and letxΔ = n

b−a

Δx i, for each i, , ..,xi=a+i=a+n

b−a = 0 1 . n

Choose sample points . A Riemann Sum is any expression of the form ε [x, ]xi*i−1 xi

(x)Δx∑

n

i=1

fi*

We say that the function is Riemann Integrable on [a,b] if converges tof(x)Δxlim

n→∞ ∑

n

i=1

fi*

the same value for every choice of sample points. In such a case the definite integral of

f from a to b is:

(x)dx (x)Δx

∫

b

a

f= lim

n→∞ ∑

n

i=1

fi

Theorem

● If f is continuous on [a,b], or has only a finite number of jump discontinuities, then

f is Riemann integrable on [a,b]

Net area under a curve

● If , then = area under the curve (x)f≥ 0 (x)dx

∫

b

a

f(x)y=f

● If , then is negative

the area between the curve and the x-axis(x)f< 0 (x)dx

∫

b

a

f< 0

● In general difference between the areas above and below the x-axis(x)dx

∫

b

a

f=

Example

1. represents the area of quarter circle of radius 3, hencedx

∫

3

0

√9 − x2

dx π π

∫

3

0

√9 − x2=4

1· 32=4

9

2. If f(x) = {x, x<2; 5-x, x 2 then the integral can be computed by≥ (x)dx

∫

4

−1

f

summing/subtracting the areas shown below:

###### You're Reading a Preview

Unlock to view full version

Subscribers Only