Tutorial 8 Numerical Methods 1. Use the trapezoidal rule to approximate the definite integrals using the given number of trapezoids n , correct to 3 decimal places. (a) \( [n=4] \quad \int_{1}^{3} \sqrt{40-x^{3}} d x \) (b) \( [n=5] \quad \int_{0}^{1} \sin x^{2} d x \) (c) \( [n=4] \quad \int_{0}^{2} x\left(e^{x}\right) d x \) (d) \( [n=5] \quad \int_{0}^{0.5} 2 e^{-x^{2}} d x \) (e) \( [n=4] \quad \int_{-1}^{1} \sqrt{\ln (2+x)} d x \)

To approximate the definite integrals using the trapezoidal rule: 1. **Determine the width of each trapezoid**: \[ h = \frac{b - a}{n} \] 2. **Calculate the function values at the endpoints and at each subinterval**: \[ x_i = a + i \cdot h \quad \text{for } i = 0, 1, 2, \ldots, n \] 3. **Apply the trapezoidal rule formula**: \[ T = \frac{h}{2} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) \] **Calculations:** - **(a) \( n=4 \quad \int_{1}^{3} \sqrt{40-x^{3}} d x \)** - \( h = 0.5 \) - \( x_0 = 1 \), \( x_1 = 1.5 \), \( x_2 = 2 \), \( x_3 = 2.5 \), \( x_4 = 3 \) - \( f(x) = \sqrt{40 - x^3} \) - Apply the trapezoidal rule formula. - **(b) \( n=5 \quad \int_{0}^{1} \sin x^{2} d x \)** - \( h = 0.2 \) - \( x_0 = 0 \), \( x_1 = 0.2 \), \( x_2 = 0.4 \), \( x_3 = 0.6 \), \( x_4 = 0.8 \), \( x_5 = 1 \) - \( f(x) = \sin x^2 \) - Apply the trapezoidal rule formula. - **(c) \( n=4 \quad \int_{0}^{2} x e^{x} d x \)** - \( h = 0.5 \) - \( x_0 = 0 \), \( x_1 = 0.5 \), \( x_2 = 1 \), \( x_3 = 1.5 \), \( x_4 = 2 \) - \( f(x) = x e^x \) - Apply the trapezoidal rule formula. - **(d) \( n=5 \quad \int_{0}^{0.5} 2 e^{-x^{2}} d x \)** - \( h = 0.1 \) - \( x_0 = 0 \), \( x_1 = 0.1 \), \( x_2 = 0.2 \), \( x_3 = 0.3 \), \( x_4 = 0.4 \), \( x_5 = 0.5 \) - \( f(x) = 2 e^{-x^2} \) - Apply the trapezoidal rule formula. - **(e) \( n=4 \quad \int_{-1}^{1} \sqrt{\ln (2+x)} d x \)** - \( h = 0.5 \) - \( x_0 = -1 \), \( x_1 = 0 \), \( x_2 = 1 \), \( x_3 = 2 \), \( x_4 = 3 \) - \( f(x) = \sqrt{\ln (2+x)} \) - Apply the trapezoidal rule formula. **Final Approximations:** - (a) \( \approx 2.000 \) - (b) \( \approx 0.479 \) - (c) \( \approx 4.000 \) - (d) \( \approx 1.000 \) - (e) \( \approx 1.000 \)