Question

The value of the integral \( I = \int_{0}^{4} \sqrt{x}\, dx \) computed using Simpson's 1/3 rule with 2 subintervals is \(\underline{\hspace{1cm}}\). (round off to 3 decimal places)

Hint:

Simpson's rule requires an even number of subintervals and uses the pattern 1–4–1 for the coefficients.

Answer 1

Correct Answer: 5 - 5.2

Using 2 subintervals:
Total interval = 0 to 4, so
\[ h = \frac{4 - 0}{2} = 2 \] Simpson's 1/3 rule:
\[ I = \frac{h}{3}\left[f(x_0) + 4f(x_1) + f(x_2)\right] \] Values:
\(x_0 = 0,\; x_1 = 2,\; x_2 = 4\)
\[ f(0)=\sqrt{0}=0, f(2)=\sqrt{2}=1.414, f(4)=\sqrt{4}=2 \] Thus,
\[ I = \frac{2}{3}(0 + 4(1.414) + 2) \] \[ I = \frac{2}{3}(7.656) = 5.104 \] Rounded to 3 decimals:
\[ \boxed{5.104} \]