Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
Lecture 4 - Computer Science Department - University of Kentucky
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Two Dimensional Integration<br />
For one dimensional numerical integration on [0,1], using uniform space<br />
h = 1/n<br />
n−1<br />
1 1<br />
⎛ i ⎞<br />
∫<br />
f<br />
(<br />
x<br />
)<br />
dx<br />
≈<br />
[<br />
f<br />
(0)<br />
+<br />
2<br />
∑<br />
f<br />
⎜<br />
⎟ +<br />
f<br />
(1)]<br />
0<br />
2h<br />
⎝ n ⎠<br />
i=<br />
1<br />
n<br />
⎛ i ⎞<br />
= ∑ Ai<br />
f ⎜ ⎟<br />
i= 0<br />
⎝<br />
n<br />
⎠<br />
For two dimensional integration on a unit square<br />
1 1<br />
∫ ∫<br />
0 0<br />
f<br />
(<br />
x<br />
,<br />
y<br />
)<br />
dx<br />
dy ≈<br />
=<br />
≈<br />
1<br />
n<br />
∫ ∑<br />
0<br />
i=<br />
1<br />
n<br />
∑<br />
i<br />
=<br />
0<br />
n<br />
∑<br />
i=<br />
0<br />
= n<br />
A<br />
A<br />
n<br />
A<br />
i<br />
∑∑<br />
i= 0 j=<br />
0<br />
∫<br />
i<br />
1<br />
0<br />
n<br />
∑<br />
i<br />
j=<br />
0<br />
A<br />
⎛ i<br />
f<br />
⎜<br />
,<br />
⎝ n<br />
i<br />
f<br />
⎛ ⎜<br />
⎝<br />
A<br />
A<br />
j<br />
j<br />
i<br />
n<br />
f<br />
f<br />
⎞<br />
y<br />
⎟<br />
dy<br />
⎠<br />
⎞<br />
, y⎟<br />
dy<br />
⎠<br />
⎛<br />
⎜<br />
⎝<br />
⎛<br />
⎜<br />
⎝<br />
i<br />
n<br />
i<br />
n<br />
,<br />
,<br />
j<br />
n<br />
j<br />
n<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
⎟<br />
⎠<br />
26
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