Simpson'S Rule Worksheet With Answers Page 2
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1
2We consider the definite integral
b
f (x) dx.
a
We assume that f (x) is continuous on [a, b] and we divide [a, b] into an even number n of subintervals of
equal length
b
a
∆x =
n
using the n + 1 points
x
= a,
x
= a + ∆x,
x
= a + 2∆x,
. . . ,
x
= a + n∆x = b.
0
1
2
n
We can compute the value of f (x) at these points.
y
= f (x
),
y
= f (x
),
y
= f (x
),
. . . ,
y
= f (x
).
0
0
1
1
2
2
n
n
y
y
4
y
y
0
y
n
y
3
1
y
y
2
n 2
y
n 1
x
a = x
x
x
x
x
x
x
x
= b
0
1
2
3
4
n 2
n 1
n
∆x
We can estimate the integral by adding the areas under the parabolic arcs through three successive points.
b
∆x
∆x
∆x
f (x) dx
(y
+ 4y
+ y
) +
(y
+ 4y
+ y
) +
+
(y
+ 4y
+ y
)
0
1
2
2
3
4
n 2
n 1
n
3
3
3
a
By simplifying, we obtain Simpson’s rule formula.
b
∆x
f (x) dx
(y
+ 4y
+ 2y
+ 4y
+ 2y
+
+ 4y
+ y
)
0
1
2
3
4
n 1
n
3
a
Example. Use Simpson’s rule with n = 6 to estimate
4
3
1 + x
dx.
1
4 1
For n = 6, we have ∆x =
= 0.5. We compute the values of y
, y
, y
, . . . , y
.
0
1
2
6
6
x
1
1.5
2
2.5
3
3.5
4
3
y =
1 + x
2
4.375
3
16.625
28
43.875
65
Therefore,
4
0.5
3
1 + x
dx
2 + 4 4.375 + 2(3) + 4 16.625 + 2 28 + 4 43.875 +
65
3
1
12.871
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