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dalasina
Mar 14, 2011, 11:11 PM
Evaluate the integral
I=∫_0^2π▒〖Cos² xdx〗
Using the following methods with 6 evaluations:
Trapezoidal rule
Simpson's 1/3 rule
Simpson's 1/8 rule
Gauss-Legendre quadrature formula
Unknown008
Mar 14, 2011, 11:37 PM
Could you retype your problem without using copy paste from a document?
Something like this:
Int ^2_0 (function) dx
dalasina
Mar 15, 2011, 01:27 AM
First of all, Thank you very much for your kind reply.
The question is:
Evaluate the integral I= Int ^0_2pi ( ( Cos^2) x )dx
Using the following methods with 6 evaluations:
(a) Trapezoidal rule
(b) Simpson's 1/3 rule
(c) Simposn's 1/8 rule
(d) Gauss-Legendre quadrature formula
Thank you again.
Unknown008
Mar 15, 2011, 02:26 AM
(a) With 6 intervals, you get the trapezium height as \frac{\pi}{3}
So, find the points on the graph at pi/3 intervals.
1. 0 --> cos^2(0) = 1
2. pi/3 --> cos^2(pi/3) = 0.25
3. 2pi/3 --> cos^2(2pi/3) = 0.25
4. pi --> cos^2(pi) = 1
5. 4pi/3 --> cos^2(4pi/3) = 0.25
6. 5pi/3 --> cos^2(5pi/3) = 0.25
7. 2pi --> cos^2(2pi) = 0.25
Then, evaluate, with 6 intervals;
\int^{b}_a f(x) dx \approx \frac12 \(d)(y_6 + y_0 + 2(y_1+y_2+y_3+y_4+y_5))
\int^{2\pi}_0 \cos^2\ x\ dx \approx \frac12 \(\frac{\pi}{3}\)(1 + 0.25 + 2(0.25+0.25+1+0.25+0.25))
Sorry, for the others, I don't know those approximations... :(