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:

For the trapezoidal rule, divide the range into equal segments and use the values of the function at the endpoints and the midpoints of the segments. Multiply by $\frac{h}{2}$ to get the approximation.

Answer 1

The Correct Option is C

We will use the trapezoidal rule with two equal segments to estimate the value of the integral $\int_2^4 x^2 \, dx$. The trapezoidal rule for $n$ segments is given by the formula: \[ T_n = \frac{h}{2} \left[ f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right] \] where $h = \frac{b-a}{n}$, $a$ is the lower limit, $b$ is the upper limit, and $f(x)$ is the function being integrated.
Step 1: Apply the trapezoidal rule for two segments.
For this case, $a = 2$, $b = 4$, and $n = 2$:
- The width of each segment, $h = \frac{4-2}{2} = 1$.
- The function is $f(x) = x^2$.
Step 2: Evaluate the function at the limits and the midpoint:
- $f(2) = 2^2 = 4$
- $f(4) = 4^2 = 16$
- $f(3) = 3^2 = 9$ (midpoint)
Step 3: Calculate the trapezoidal approximation: \[ T_2 = \frac{1}{2} \left[ f(2) + 2f(3) + f(4) \right] \] \[ T_2 = \frac{1}{2} \left[ 4 + 2 \times 9 + 16 \right] = \frac{1}{2} \left[ 4 + 18 + 16 \right] = \frac{1}{2} \times 38 = 19.0 \] Thus, the estimated value of the integral is $19.0$.