MATH 250a: Fall Semester 2007 - Integration: Examples and Improper Integrals, Study notes of Differential Equations

Download MATH 250a: Fall Semester 2007 - Integration: Examples and Improper Integrals and more Study notes Differential Equations in PDF only on Docsity!

MATH 250a

Fall Semester 2007

Section 2 (J. M. Cushing)

Thursday, October 4

http://math.arizona.edu/~cushing/250a.html

Chapter 7: Integration

Sections 1 – 6

Two main themes:

Calculation of anti-derivatives (integrals)

One goal is to evaluate integrals usingthe Fundamental Theorem of Calculus

Numerical approximation of integrals

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫

decreasing

( ) 2 2 2 2 2

x x

dy

xe

dx dy

x

e

dx

− −

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫

decreasing

LEFT(

n

) & RIGHT(

n

provide error bounds

( ) 2 2 2 2 2

x x

dy

xe

dx dy

x

e

dx

− −

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫

concavity changes at

TRAP(

n

) & MID(

n

provide no error bounds

x

( ) 2 2 2 2 2

x x

dy

xe

dx dy

x

e

dx

− −

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫ n

RIGHT(

n

LEFT(

n 2

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫ n

RIGHT(

n

LEFT(

n 2

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫ n

RIGHT(

n

LEFT(

n 2

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫ n

RIGHT(

n

LEFT(

n 2

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0 x e dx − ∫ n

TRAP(

n

MID(

n 2

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** n

TRAP(

n

MID(

n 2

2 1 0 x e dx − ∫

EXAMPLE

y = e

-x 2 x

**-

0 1 2 3 1.0 0.8 0.6 0.4 0.** 2 1 0

x e dx − ≈ ∫ n

TRAP(

n

MID(

n 2

n

RIGHT(

n

LEFT(

n

n

TRAP(

n

MID(

n 2

Assuming 0.7468 is the correct answer, note that

TRAP & MID provide more accuracy for each

n

TRAP & MID

converge faster

as

n

increases

n

LEFT(

n

ERROR(

n

x

• 4

x

• 8

x

• 16

x

• 32

x

• 64

x

• 128

x

• 256

x

• 512

x

n

TRAP(

n

ERROR(

n 2

x

• 4

x

• 8

x

• 16

x

• 32

Assuming 0.7468 is the correct answer, note that

TRAP & MID provide more accuracy at each step

TRAP & MID converge faster as

n

increases