Numerical
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The trapezoidal method splits the area under a curve into trapezoids, calculates the area of each trapezoid, and sums the individual areas to approximate the total area under the curve, which represents the definite integral. It uses the formula: Integral = (h/2) * (y0 + 2*y1 + 2*y2 + ... + 2*yn-1 + yn), where h is the width of each interval between the x-values x0, x1, etc., and y0, y1, etc. are the corresponding y-values of the function. Simpson's 1/3 Rule similarly divides the interval into sub-intervals, but approximates each using a quadratic curve
![Example:
• ⌠1
4 (x2 +2X)dx with n=6
• ANSWER:
• n=6, h=(b-a)/n=3/6=0.5
•
• ⌠1
4 (x2 +2X)dx=h/2[(3+24)+ 2*(5.25+8+11.25+15+19.25)]=36.125
•
X 1 1.5 2 2.5 3 3.5 4
Y 3 5.25 8 11.25 15 19.25 24](https://image.slidesharecdn.com/numerical-210503041806/85/Numerical-4-320.jpg)
Numerical
- 2. The basic working principle of the trapezoidal method c program is that trapezoidal method splits the area under the curve into a number of trapeziums. Then, area of each trapezium is calculated, and all the individual areas are summed up to give the total area under the curve which is the value of definite integral. In the figure, area under the curve between the points x0 and xn is to be determined. In order to determine the area, we divide the total interval (xn– x0) into ‘n’ small interval each of length ‘h’: h=(xn– x0)/n After that, the C source code for trapezoidal method uses the following formula to calculate the value of definite integral:
- 3. Steps • 1. take n that is number of intervals • 2. calculate h=(b-a)/n , a is lower limit and b is upper limit • 3. find x0=a, x1=a+h, x2=a+2.h , ….. • 4. using y=f(x), Find y0,y1,y2…. • 5.
- 4. Example: • ⌠1 4 (x2 +2X)dx with n=6 • ANSWER: • n=6, h=(b-a)/n=3/6=0.5 • • ⌠1 4 (x2 +2X)dx=h/2[(3+24)+ 2*(5.25+8+11.25+15+19.25)]=36.125 • X 1 1.5 2 2.5 3 3.5 4 Y 3 5.25 8 11.25 15 19.25 24
- 5. Simpson’s 1/3rd Rule: • Simpson’s Rule is a Numerical technique to find the definite integral of a function within a given interval. • The function is divided into many sub-intervals and each interval is approximated by a quadratic curve. And the area is then calculated to find the integral. The more is the number of sub-intervals used, the better is the approximation. • NOTE: The no. of sub-intervals should be EVEN.
- 6. Integral = *((y0 + yn ) +4(y1 + y3 + ……….+ yn-1 ) + 2(y2 + y4 +……….+ yn-2 ))