Summary Numeric Integration | Calculus II Notesq

1.11 Numeric Integration
Saf(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
=