Simpson'S Rule Worksheet With Answers
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1
2Simpson’s rule is a numerical method that approximates the value of a definite integral by using quadratic
polynomials.
2
Let’s first derive a formula for the area under a parabola of equation y = ax
+ bx + c passing through the
three points: ( h, y
), (0, y
), (h, y
).
0
1
2
y
2
y = ax
+ bx + c
(0, y
)
1
(h, y
)
2
( h, y
)
0
y
1
y
2
y
0
∆x
x
0
h
h
h
2
A =
(ax
+ bx + c) dx
h
h
3
2
ax
bx
=
+
+ cx
3
2
h
3
2ah
=
+ 2ch
3
h
2
=
2ah
+ 6c
3
2
Since the points ( h, y
), (0, y
), (h, y
) are on the parabola, they satisfy y = ax
+ bx + c. Therefore,
0
1
2
2
y
= ah
bh + c
0
y
= c
1
2
y
= ah
+ bh + c
2
Observe that
2
2
2
y
+ 4y
+ y
= (ah
bh + c) + 4c + (ah
+ bh + c) = 2ah
+ 6c.
0
1
2
Therefore, the area under the parabola is
h
∆x
A =
(y
+ 4y
+ y
) =
(y
+ 4y
+ y
) .
0
1
2
0
1
2
3
3
Gilles Cazelais. Typeset with L A T E X on April 23, 2008.
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