Trapezoidal Rule - GATE MA Engineering Mathematics Free MCQ Test with solutions

The following table shows the different methods of numerical integration and degree of polynomials for which they will produce results of minimum error or zero error:

From the above table, it is clear that both Trapezoidal Rule polynomials of degree ≤ 1

The trapezoidal method is based on the assumption that the mid-area is the mean of the end areas. It is true only if the prismoid is composed of prisms and wedges only but not of pyramids.

The Trapezoidal Rule for approximating  is given by

Calculation:


Hence the correct answer is 9.045.

This method is based on the assumption that the mid-area is the mean of the end areas, which make it the Average end area method.

Trapezoidal Rule:

• The geometric significance of this rule is that the curve y = f(x) us replaced by n straight lines joining the point (x_o, y_o) and (x₁, y₁); (x₁, y₁) and (x₂, y₂); ... ; (x_n-1, y_n-1) and (x_n, y_n).
• The area bounded by the curve is given by - 
  

Here,
h = x_n - x_n-1
Hence,
The area under a straight line is an estimate of the integral of f(x) between the limits a and b will be given by -

Trapezoidal formula is based on the assumption that the mid-area is the mean of the end area. Based on this, the trapezoidal formula will be worked out and further calculations are done.

Volume of the third section of a prismoid can be calculated as,
V = d/2 (A3 + A4). On substitution, we get
V = 4/2 (76.32 + 24.56)
V = 201.76 cu. m.