PPT: Numerical Integration | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


WHAT IS NUMERICAL INTEGRATION?
• Approximate computation of a definite integral using numerical 
techniques
• Tabulated at regularly spaced intervals 
• Gives the approximate calculation
• Different from analytical integration 
Page 2


WHAT IS NUMERICAL INTEGRATION?
• Approximate computation of a definite integral using numerical 
techniques
• Tabulated at regularly spaced intervals 
• Gives the approximate calculation
• Different from analytical integration 
NUMERICAL INTEGRATION
TRAPEZOIDAL 
RULE
SIMPSON’S RULE
GAUSSIAN 
QUADRATURE
Page 3


WHAT IS NUMERICAL INTEGRATION?
• Approximate computation of a definite integral using numerical 
techniques
• Tabulated at regularly spaced intervals 
• Gives the approximate calculation
• Different from analytical integration 
NUMERICAL INTEGRATION
TRAPEZOIDAL 
RULE
SIMPSON’S RULE
GAUSSIAN 
QUADRATURE
TRAPEZOIDAL RULE
• Fundamental method of Numerical Integration
• Trapezoids are used for finding the area under curve
• Trapezoids better fits the curve, less error
Page 4


WHAT IS NUMERICAL INTEGRATION?
• Approximate computation of a definite integral using numerical 
techniques
• Tabulated at regularly spaced intervals 
• Gives the approximate calculation
• Different from analytical integration 
NUMERICAL INTEGRATION
TRAPEZOIDAL 
RULE
SIMPSON’S RULE
GAUSSIAN 
QUADRATURE
TRAPEZOIDAL RULE
• Fundamental method of Numerical Integration
• Trapezoids are used for finding the area under curve
• Trapezoids better fits the curve, less error
• Evaluating, I= 
?? ?? ?? ?? ???? by Trapezoidal Rule
• Consider an expanded view of a general region
• Area of each trapezoid be,
 
?? ??-1
?? ?? ?? ?? ???? ˜
?? ??-1
+?? ?? 2
(??? )
And,
 
?? ?? ?? ??+1
?? ?? ???? ˜
?? ?? +?? ??+1
2
(??? )
• The integral over two panel is given by,
 
?? ??-1
?? ??+1
?? ?? ???? ˜
?? ?? +?? ??+1
2
??? +
?? ??-1
+?? ?? 2
??? =
??? 2
(?? ??-1
+2?? ?? +?? ??+1
)
• The Trapezoidal Rule approximation to a integral over the entire 
Interval is ,
 
?? ?? ?? ?? ???? ˜
??? 2
{ ?? 0
+?? ?? )+2(?? 1
+?? 2
+?+?? ?? -1
}
Where ?? ?? =?? ?? ?????? ?? ?? =?? ?? ?x = 
?? -?? ??
Page 5


WHAT IS NUMERICAL INTEGRATION?
• Approximate computation of a definite integral using numerical 
techniques
• Tabulated at regularly spaced intervals 
• Gives the approximate calculation
• Different from analytical integration 
NUMERICAL INTEGRATION
TRAPEZOIDAL 
RULE
SIMPSON’S RULE
GAUSSIAN 
QUADRATURE
TRAPEZOIDAL RULE
• Fundamental method of Numerical Integration
• Trapezoids are used for finding the area under curve
• Trapezoids better fits the curve, less error
• Evaluating, I= 
?? ?? ?? ?? ???? by Trapezoidal Rule
• Consider an expanded view of a general region
• Area of each trapezoid be,
 
?? ??-1
?? ?? ?? ?? ???? ˜
?? ??-1
+?? ?? 2
(??? )
And,
 
?? ?? ?? ??+1
?? ?? ???? ˜
?? ?? +?? ??+1
2
(??? )
• The integral over two panel is given by,
 
?? ??-1
?? ??+1
?? ?? ???? ˜
?? ?? +?? ??+1
2
??? +
?? ??-1
+?? ?? 2
??? =
??? 2
(?? ??-1
+2?? ?? +?? ??+1
)
• The Trapezoidal Rule approximation to a integral over the entire 
Interval is ,
 
?? ?? ?? ?? ???? ˜
??? 2
{ ?? 0
+?? ?? )+2(?? 1
+?? 2
+?+?? ?? -1
}
Where ?? ?? =?? ?? ?????? ?? ?? =?? ?? ?x = 
?? -?? ?? SIMPSON’S 1/3 RULE
• A further improvement over Trapezoidal rule is Simpson’s Rule
• Based on use of parabolic arcs (quadratic function) to approximate the 
curve instead of the straight lines employed in the trapezoid rule
• Connecting 3 points, the number of interval has
to be even