To find the approximate value of the integral \(\int_1^2 e^{-x/2} \, dx\) using Simpson's \(1/3\)rd rule with four intervals, we follow these steps:
- Simpson's \(1/3\)rd rule requires an even number of intervals, so using four intervals is appropriate.
- To apply Simpson's Rule, divide the interval \([1, 2]\) into \(4\) subintervals. The width of each subinterval, \(h\), is calculated as: \(h = \frac{b-a}{n} = \frac{2-1}{4} = 0.25\)
- Compute the values of \(x_i\) at each point:
- \(x_0 = 1\)
- \(x_1 = 1 + 0.25 = 1.25\)
- \(x_2 = 1 + 2 \times 0.25 = 1.5\)
- \(x_3 = 1 + 3 \times 0.25 = 1.75\)
- \(x_4 = 2\)
- Evaluate \(f(x_i) = e^{-x_i/2}\) for each point:
- \(f(x_0) = e^{-1/2} \approx 0.6065\)
- \(f(x_1) = e^{-1.25/2} \approx 0.4724\)
- \(f(x_2) = e^{-1.5/2} \approx 0.3679\)
- \(f(x_3) = e^{-1.75/2} \approx 0.2865\)
- \(f(x_4) = e^{-2/2} \approx 0.2431\)
- Apply Simpson's \(1/3\)rd rule:
\(\int_1^2 e^{-x/2} \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \left( f(x_1) + f(x_3) \right) + 2 f(x_2) + f(x_4) \right]\)
Substitute the values:
\(\int_1^2 e^{-x/2} \, dx \approx \frac{0.25}{3} \left[ 0.6065 + 4(0.4724 + 0.2865) + 2(0.3679) + 0.2431 \right]\)
\(= \frac{0.25}{3} \left[ 0.6065 + 4 \times 0.7589 + 0.7358 + 0.2431 \right]\)
\(= \frac{0.25}{3} \left[ 0.6065 + 3.0356 + 0.7358 + 0.2431 \right]\)
\(= \frac{0.25}{3} \times 4.6210 \approx 0.477\)
The approximate value of the integral is \(\boxed{0.477}\).