Question

Evaluation of the integral \(\int_{2}^{4} x^2 \, dx\) using the trapezoidal rule (with two equal segments) gives a value of .......

• A. 14.5
• B. 22.5
• C. 19.0
• D. 18.6

Hint:

The trapezoidal rule divides the integration interval into subintervals and approximates the area under the curve using trapezoids. For \( n = 2 \), calculate at three points: start, midpoint, and end.

Answer 1

The Correct Option is C

The trapezoidal rule for integral approximation over \([a, b]\) using \(n = 2\) segments is given by: \[ \int_{a}^{b} f(x)\, dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + f(x_2) \right] \] where: \[ h = \frac{b - a}{n} = \frac{4 - 2}{2} = 1 \] \[ x_0 = 2, x_1 = 3, x_2 = 4 \] \[ f(x) = x^2 ⇒ f(2) = 4, f(3) = 9, f(4) = 16 \] \[ ⇒ \int_{2}^{4} x^2 dx \approx \frac{1}{2} \left[ 4 + 2 \times 9 + 16 \right] = \frac{1}{2}(4 + 18 + 16) = \frac{38}{2} = 19.0 \]