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Summary Numeric Integration | Calculus II Notesq
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Numeric approximation for an integral by using the midpoint rule, trapezoidal rule, and simpson's rule. Includes examples with diagrams and explanations.
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1.11 Numeric IntegrationSaf(x)dx=hflxx*) Ax (Rn- Riemann Sum)
...
for any n, Rn is a numeric approximation for Saf(x)dX
midpointrule approximation:
Saf(x)dx = (f(xi) f(x) + ..
+
.
+
f(xn)] ax
!
X, A AX. X=b
Ax=
b,9, No 9 + &n=a +2AX... Xn-i-b-AX,
=
=
x, X, Xar= n=Xn-1
XentWars,
x, + xn
x0X,, x.
=
+
=
. . .
2 2
xj - 1 X,
example:So i 4.ex
approximate the integral using the midpointrule, n 8
=
firstsetupall needed x-values
I 6
Ax =
=
Xo 0,x,
y,x 8....,x= 8x8 8
= = =
=
= =
x,
x B,xs y, ...,xo
I
=
= =
=
apply the equation
So,
i?..*
Y
w 1 x,
+
I **
=3.1429
.....
The trapezoidal rule: -...-l
A (r
=
e)w/2
+
SYf(x)dx kf(x) f(x) f(x) .
=
+ + +
. .
+
f(xn -) kf(xn)
+
:Xo a x, A AX, X1 a 2AX, Xn-, b-AX, Xn=b
=
=
+ + =
=
...,
example:So, *.dx, n8 =
x0 0, x,
8.x 2... Xo
8 1
= = =
=
=
using the trap ruleequation:
So,Yadx it*:"... its
3.139
=
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