Question

Given \( \int_{-3}^{3} (x^5 + 4x^3) dx \), the integral value using Trapezoidal rule is:

• A. \( 0 \)
• B. \( 81 \)
• C. \( -1 \)
• D. \( 18 \)

Hint:

For any symmetric interval, the Trapezoidal rule, Simpson's rule, and the actual integral will all yield zero if the function is odd.

Answer 1

The Correct Option is A

Concept: The Trapezoidal rule approximates the area under a curve by dividing it into trapezoids.

• Trapezoidal Rule: \( T = \frac{h}{2} [f(x_0) + 2(f(x_1) + \dots) + f(x_n)] \).

• Properties of Odd Functions: An odd function satisfies \( f(-x) = -f(x) \).

• The integral of an odd function over a symmetric interval \( [-a, a] \) is zero.

Step 1: Analyze the function \( f(x) \).
Let \( f(x) = x^5 + 4x^3 \). Test for parity: \[ f(-x) = (-x)^5 + 4(-x)^3 = -x^5 - 4x^3 = -f(x) \] The function is an odd function.

Step 2: Check for symmetry in the interval.
The interval is from \( -3 \) to \( 3 \). For any chosen step size \( h \) and symmetric points \( x_i \) and \( -x_i \): \[ f(x_i) + f(-x_i) = f(x_i) - f(x_i) = 0 \]

Step 3: Apply the Trapezoidal Rule.
The rule uses the sum of the function values. Because every positive value at \( x \) is cancelled by a negative value at \( -x \): \[ \text{Sum of terms} = 0 \] Final Answer: (A)