Resolucion De Integrales De Superficie Usando El Método De Simpson

Double Integrals

Consider the double integral

22 - Multiple Integrals

R

Per-Olof Persson

f (x, y ) dA,

R = {(x, y ) | a ≤ x ≤ b, c ≤ y ≤ d}Partition [a, b] and [c, d] into even number of subintervals n, m

persson@berkeley.edu

Step sizes h = (b − a)/n and k = (d − c)/m

Department ofMathematics
University of California, Berkeley

Write the integral as an iterated integral

Math 128A Numerical Analysis

R

f (x, y ) dA =

b
a

d
c

f(x, y ) dy

dx

and use any quadrature rule in an iterated manner.

Composite Simpson’s Rule Double Integration

Non-Rectangular Regions

The CompositeSimpson’s rule gives
b

d

a

c

f (x, y ) dy

dx =

hk
9

n

m

wi,j f (xi , yi ) + E
i=0 i=0

where xi = a + ih, yi = c + jk , wi,j arethe products of the nested
Composite Simpson’s rule coefficients (see below), and the error is
E=−

(d − c)(b − a) 4 ∂ 4 f
∂4f
h
(¯, µ) + k 4 4 (ˆ, µ)
η¯
ηˆ4
180
∂x
∂y
d1

4

2

4

16

8

16

4

c1
a

4

2

4

1
b

Gaussian Double Integration

For Guassian integration, firsttransform the roots rn,j from
[−1, 1] to [a, b] and [c, d], respectively
The integral is then
b
a

d
c

f (x, y ) dy dx ≈

(b − a)(d − c)
4

n

b
ad(x)
c(x)

n

cn,i cn,j f (xi , yi )
i=1 j =1

Similar techniques can be used for non-rectangular regions

f (x, y ) dy dx

The step size for x is stillh = (b − a)/n, but for y it varies with x:
k (x) =

1

4

The same technique can be applied to double integrals of the form

d(x) − c(x)
m