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

Step 1: Find the exact value using the antiderivative.
For \( f(x) = x^5+4x^3 \), an antiderivative is \( F(x) = \dfrac{x^6}{6} + x^4 \). Both terms involve only even powers of \( x \), so \( F(-x) = F(x) \) for every \( x \).

Step 2: Evaluate at the endpoints.
\[ \int_{-3}^{3}(x^5+4x^3)\,dx = F(3)-F(-3) \] Since \( F(3) = F(-3) \), this difference is exactly zero.

Step 3: Relate this to the trapezoidal estimate.
The trapezoidal rule uses a symmetric grid of points around zero. Because \( f \) is odd, the function value at every node \( x_i \) is cancelled by the value at \( -x_i \), so the weighted sum used by the rule also collapses to zero, matching the true integral. \[ \boxed{0} \]