Numerical Integration - Numerical Methods in Engineering with MATLAB - page 241

which yieldsfor the Jacobian matrix

⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

∂ x

∂ξ

∂ y

∂ξ

1 + η

4

1

J ( ξ,η ) =

∂ x

∂η

∂ y

∂η

5 + ξ

4

0

Thus the area scale factoris

+ ξ
4
Nowwecan map the integralfrom the quadrilateral to the standard rectangle.Refer-
ring to Eq. (6.45), weobtain

5

| ξ,η | =

J ( )

1 1 (1 5

(5 + ξ )(1 + η ) + ξ

4

) 2 I = + ξ + d ξ d η

4

− 1 − 1

1 1 45

16 +

d

21
8 ξ +

29
16 ξ

1
4 ξ

25
16 η +

5
8 ξη +

1
16 ξ

2 3 2

= + + η ξ d η

− 1 − 1

Noting thatonly evenpowersof ξ and η contribute to the integral, wecan simplify the

integralto

1 1 45

16 +

2 d

29
16 ξ

41
3

I = ξ d η =

− 1 − 1

EXAMPLE 6.14
Evaluate the integral

1 1

cos π

x
2

cos π

y
2

dx dy

− 1 − 1

by Gauss-Legendrequadratureoforder three.

Solution From the quadratureformulainEq. (6.40), wehave

3 3

A i A j cos π cos π

x i
2

y j
2

I =

i = 1 j = 1

y

1

a a

b

b b

0 x

a b a

−1

−1 1

0

The integrationpoints are shown in the figure; their coordinates and the correspond-
ing weights arelistedinTable 6.3. Note that the integrand, the integrationpoints and