Q.60 ∫_-1¹ f(x) dx calculated using trapezoidal rule for the values given in the table is ________ (rounded off to 2 decimal places).

x	-1	-2/3	-1/3	0	1/3	2/3	1
f(x)	0.37	0.51	0.71	1.0	1.40	1.95	2.71

 

Given Data

x	-1	-2/3	-1/3	0	1/3	2/3	1
f(x)	0.37	0.51	0.71	1.0	1.40	1.95	2.71

Step Size

h = (1 – (-1)) / 6 = 2/6 = 1/3

Trapezoidal Rule Formula

∫_a^b f(x)dx ≈ h × [ (f₀ + f_n) / 2 + Σ f_i ]

Step-by-Step Calculation

1. Average of Endpoints

f(-1) = 0.37,   f(1) = 2.71
(f₀ + f_n) / 2 = (0.37 + 2.71) / 2 = 1.54

2. Sum of Interior Values

0.51 + 0.71 + 1.0 + 1.40 + 1.95 = 5.57

3. Add Terms

Total = 1.54 + 5.57 = 7.11

4. Multiply by h

∫_-1¹ f(x)dx ≈ (1/3) × 7.11 = 2.37

Final Answer

Integral value = 2.37 (rounded to 2 decimal places)

Interpretation

• Because values are tabulated, numerical integration is required
• Equal spacing allows trapezoidal rule application
• Simpson’s rule not applicable due to odd number of segments

Exam Tip

For equally spaced x-values, use trapezoidal rule when Simpson’s rule cannot apply.