Trapezoidal Rule Vs Simpson’s Rule In Surveying: Difference & Example

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Trapezoidal and Simpson’s rule are both  numerical methods used in approximating values of a definite integrals.

Trapezoidal rule

In this rule, the boundaries between the ends of ordinates are assumed to be straight. Thus the areas enclosed between the baseline and the irregular boundary lines are considered trapezoids.

What You Need To Know About Trapezoidal Rule/Trapezium Rule

  1. The result obtained by the trapezoidal rule is not affected because the boundary between the ordinates is considered straight.
  2. Trapezoidal rule can also be referred to as Trapezium rule.
  3. Trapezoidal rule gives an estimated result. For example, the area of an irregular piece of land obtained by trapezoidal rule is just an approximate value and not the accurate value.
  4. In trapezoidal rule, the boundary between the ordinates is considered straight.
  5. There is no limitation for this rule. This rule can be applied for any number of ordinates.
  6. Computations involved in Trapezoidal rule are not as complex as those in Simpson’s rule.
  7. Trapezoidal rule can be stated as follow:  To the sum of the first and last ordinate, twice the sum of intermediate ordinate is added. This total sum is multiplied by the common distance. Half of this product is the required area.
Formula

Simpson’s Rule

Simpson’s rule is a numerical approach to finding definite integrals where no other method is possible. The value of a definite integral is approximated using quadratic function. In this rule, the boundaries between the ends of ordinates are assumed to form an arc of parabola. Hence Simpson’s rule is sometimes referred to as parabolic rule. The results obtained by using Simpson’s rule have a high degree of accuracy and therefore, it is only used when great accuracy is required.

Simpson’s Rule/Parabolic Rule

  1. The result obtained by the Simpson’s rule is greater or lesser as the curve of the boundary is convex or concave towards the baseline.
  2. Simpson’s Rule can also be referred to as Parabolic Rule.
  3. Simpson’s rule gives accurate result when compared to Simpsons rule.
  4. In Simpson’s rule, the boundary between the ordinates is considered to be an arc of a parabola.
  5. This rule is applicable only when the number of divisions is even i.e the number of ordinates is odd.
  6. Greater computational effort is involved and rounding errors may become a more significant problem.
  7. Simpson’s rule may be stated as follow:  To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added. This total sum is multiplied by the common distance. One third of this product is the required area.

Difference Between Trapezoidal Rule And Simpson’s Rule In Tabular Form

BASIS OF COMPARISON TRAPEZOIDAL RULE SIMPSON’S RULE
Result Obtained The result obtained by the trapezoidal rule is not affected because the boundary between the ordinates is considered straight.   The result obtained by the Simpson’s rule is greater or lesser as the curve of the boundary is convex or concave towards the baseline.  
Alternative Name Trapezoidal rule can also be referred to as Trapezium rule.   Simpson’s Rule can also be referred to as Parabolic Rule.  
Accuracy Gives an estimated result. For example, the area of an irregular piece of land obtained by trapezoidal rule is just an approximate value and not the accurate value.   Gives accurate result when compared to Simpsons rule.  
Boundary Between Ordinates The boundary between the ordinates is considered straight.   The boundary between the ordinates is considered to be an arc of a parabola.  
Applicability There is no limitation for this rule. This rule can be applied for any number of ordinates.   This rule is applicable only when the number of divisions is even i.e the number of ordinates is odd.  
Computation Computations involved in Trapezoidal rule are not as complex as those in Simpson’s rule.   Greater computational effort is involved and rounding errors may become a more significant problem.  
Rule To the sum of the first and last ordinate, twice the sum of intermediate ordinate is added. This total sum is multiplied by the common distance. Half of this product is the required area.   To the sum of the first and the last ordinate, four times the sum of even ordinates and twice the sum of the remaining odd ordinates are added. This total sum is multiplied by the common distance. One third of this product is the required area.  
Formula