Question

Using single step trapezoidal rule, the value of
\[ \int_{0}^{0.5} (1+16x-20x^2)\,dx \]
is ______ (rounded off to two decimal places).

Hint:

Apply (b-a)/2 times the sum of the function values at the two endpoints of the interval.

Answer 1

Correct Answer: 1.25

Step 1: State the single-step trapezoidal rule.
\[ \int_{a}^{b} f(x)\,dx \approx \frac{b-a}{2}\left[f(a)+f(b)\right] \]

Step 2: Identify the function and limits.
\(f(x) = 1+16x-20x^2\), \(a=0\), \(b=0.5\).

Step 3: Evaluate the function at the two endpoints.
\(f(0) = 1\)
\(f(0.5) = 1+8-5 = 4\)

Step 4: Substitute into the trapezoidal formula.
\[ \int_{0}^{0.5}(1+16x-20x^2)\,dx \approx \frac{0.5}{2}\left[1+4\right] = 0.25 \times 5 = 1.25 \]

\[ \boxed{1.25} \]