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

Alternate method: geometric trapezoid area, with exact-integral cross-check.
$y_0 = f(0) = 1$, $y_1 = f(0.5) = 4$. Area $= h(y_0+y_1)/2 = 0.5(1+4)/2 = 1.25$.
Exact integral: $\left[x+8x^2-\frac{20x^3}{3}\right]_0^{0.5} = 1.667$. The trapezoidal estimate underestimates since the curve is concave down, but the question asks specifically for the single-step trapezoidal estimate, $1.25$.