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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
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FAQs on PPT: Numerical Integration - Engineering Mathematics - Civil Engineering (CE)

1. What is numerical integration?
Ans. Numerical integration is a method used to approximate the definite integral of a function by dividing the interval into smaller subintervals and calculating the area under the curve using numerical techniques.
2. Why is numerical integration important in mathematics and engineering?
Ans. Numerical integration plays a crucial role in mathematics and engineering as it allows us to numerically compute the values of definite integrals, which are often difficult or impossible to solve analytically. It is particularly useful in solving real-world problems that involve complex functions and irregular domains.
3. What are the different methods of numerical integration?
Ans. There are several methods of numerical integration, including the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods vary in their accuracy and computational complexity and are chosen based on the specific problem at hand.
4. How does the trapezoidal rule work in numerical integration?
Ans. The trapezoidal rule approximates the area under the curve by dividing the interval into trapezoids and summing their areas. It uses linear interpolation between adjacent points to estimate the function values and then calculates the sum of these areas to obtain the approximate integral.
5. What is the error in numerical integration methods?
Ans. The error in numerical integration methods depends on several factors, such as the chosen method, the number of subintervals, and the smoothness of the function being integrated. Generally, as the number of subintervals increases, the error decreases. More advanced methods, like Gaussian quadrature, can achieve higher accuracy compared to simpler methods like the trapezoidal rule.
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