Question

Using Trapezoidal rule with step size of 0.1, the value of the integral \[ \int_{0}^{0.3} f(x)\,dx \]

• A. \(6.5\)
• B. \(5.5\)
• C. \(7\)
• D. \(8\)

Hint:

Trapezoidal rule shortcut: \[ \boxed{ \text{Area} = \frac{h}{2}\left[y_0 + y_n + 2\sum y_{\text{middle}}\right] } \] Always double-check arithmetic carefully.

Answer 1

The Correct Option is C

Concept: Trapezoidal rule: \[ \int_a^b f(x)\,dx \approx \frac{h}{2}\left[y_0 + 2(y_1+y_2+\cdots+y_{n-1}) + y_n\right] \] Here, \[ h = 0.1 \] \[ y_0=0,\; y_1=20,\; y_2=30,\; y_3=60 \]

Step 1: Apply formula.
\[ \int_0^{0.3} f(x)\,dx = \frac{0.1}{2}\left[0 + 2(20+30) + 60\right] \]

Step 2: Simplify.
\[ = 0.05 \left[0 + 2(50) + 60\right] \] \[ = 0.05 (100 + 60) \] \[ = 0.05 \times 160 \] \[ = 8 \]

Step 3: Final answer.
\[ \boxed{8} \Rightarrow Option (D) \]