Test: Simpson's 1/3rd Rule - Civil Engineering (CE) MCQ

Simpson's rule is given by -

h = width of interval / step length
y_o, y₁ ........ y_n  - ordinates corresponding to x_o, x₁ ........ x_n
Error = Exact value - approximate value
Calculation:
Given:
Since the given function is second-degree polynomial.
Simpson's 1/3 rd rule also uses a second degree polynomial for approximation.
Hence there will be no error in the result
The value of I_s and I_e will be the same and hence 
e = 100 × (I_e - I_s)/I_e = 0

The formula for Simpson’s rule can be given as Δ = (d/3)*((O₀+O₄) + 4*(O₁+O₃) + 2*(O₂+O₄)). On substitution, we get
Δ = (12/3)* ((2.25+4.46) + 4*(2.25+3.23) + 2*(1.46+4.46))
Δ = 161.88 sq. m.

The Simpson’s rule is given by,

h –Width of interval / Step length
y₀, y₁ …y_n – Ordinates corresponding to x₀, x₁, ……, x_n
Calculation:
Given:

h = 0.25

Due to the presence of curvature at the boundary whether it may be concave or convex towards the base line, the results are depended. It makes them greater than that obtained from the trapezoidal rule.

Truncation error:
The following table shows the different methods of numerical integration and degree of polynomials for which they will produce exact results (i.e. no error):

Explanation:
Given,

Here,
f(x) = x² → Quadratic polynomial.
From the above table, Simpson's rule will integrate exactly the polynomials of degree ≤ 3.
∴ Error = 0
Since,
Error = Exact - Approximate
⇒ I - J = 0
⇒ I = J

The area of the segment can be found out by using,
A = (2/3)*(O1-(O0+O2/2)). On substitution, we get
A = (2/3)*(45.76-(119.65+32.87/2))
A = -20.34 Sq. m (negative sign has no significance)
A = 20.34 sq. m.

Even though Simpson’s rule assumes that short lengths of boundary between the ordinates are parabolic arcs, this method is more accurate for the case when straights act as a parallel to each other.