Question

Let \( f(x) = |x| + |x - 1| + |x - 2|, \, x \in [-1, 2] \). Which one of the following numerical integration rules gives the exact value of the integral \[ \int_{-1}^2 f(x) \, dx? \]

• A. The Simpson’s rule
• B. The trapezoidal rule
• C. The composite Simpson’s rule by dividing \( [-1, 2] \) into 4 equal subintervals
• D. The composite trapezoidal rule by dividing \( [-1, 2] \) into 3 equal subintervals

Hint:

For piecewise linear functions, composite trapezoidal rule with sufficient subintervals ensures exact integration.

Answer 1

The Correct Option is D

Step 1: Nature of \( f(x) \). The function \( f(x) \) is piecewise linear, and the composite trapezoidal rule over 3 equal subintervals captures the exact integral. Step 2: Application of the composite rule. Dividing \( [-1, 2] \) into 3 equal parts ensures the exact value of the integral because \( f(x) \) is linear in each subinterval. Step 3: Conclusion. The correct numerical rule is \( {(4)} \).