Question

Evaluate \(\int_{1}^{5} x^2 dx\) using 4 equal intervals by trapezoidal rule and Simpson's 1/3 rule, and compute the absolute difference (round to 2 decimals).

Hint:

Simpson's rule is always more accurate than trapezoidal rule for smooth polynomial functions.

Answer 1

Correct Answer: 0.66

Step size:
\[ h = \frac{5 - 1}{4} = 1 \] Function values: \[ f(x)=x^2: 1, 4, 9, 16, 25 \] Trapezoidal Rule:
\[ T = \frac{h}{2} \left[f_0 + 2(f_1+f_2+f_3) + f_4\right] \] \[ = \frac{1}{2}(1 + 2(4+9+16) + 25) \] \[ = \frac{1}{2}(1 + 2\cdot29 + 25) = \frac{1}{2}(84) = 42 \] Simpson's Rule:
\[ S = \frac{h}{3}(f_0 + 4(f_1 + f_3) + 2f_2 + f_4) \] \[ = \frac{1}{3}(1 + 4(4+16) + 2\cdot9 + 25) \] \[ = \frac{1}{3}(1 + 80 + 18 + 25) = \frac{1}{3}(124) = 41.3333 \] Difference:
\[ |42 - 41.3333| = 0.6667 \approx 0.67. \]