Question

Numerically integrate, \( f(x) = 10x - 20x^2 \) from lower limit \( a = 0 \) to upper limit \( b = 0.5 \). Use Trapezoidal rule with five equal subdivisions. The value (in units, round off to two decimal places) obtained is \(\underline{\hspace{2cm}}\).

Hint:

The Trapezoidal rule is a method for numerical integration that approximates the integral by dividing the area under the curve into trapezoids.

Answer 1

Correct Answer: 0.38

The Trapezoidal rule for numerical integration is given by the formula: \[ I = \frac{h}{2} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right], \] where \( h = \frac{b - a}{n} \) is the width of each subdivision, and \( n = 5 \) is the number of subdivisions. The step size is: \[ h = \frac{0.5 - 0}{5} = 0.1. \] Now, applying the Trapezoidal rule: \[ I = \frac{0.1}{2} \left[ f(0) + 2 \left( f(0.1) + f(0.2) + f(0.3) + f(0.4) \right) + f(0.5) \right]. \] First, calculate the values of \( f(x) \) at each point: \[ f(0) = 10(0) - 20(0)^2 = 0, \] \[ f(0.1) = 10(0.1) - 20(0.1)^2 = 1 - 0.2 = 0.8, \] \[ f(0.2) = 10(0.2) - 20(0.2)^2 = 2 - 0.8 = 1.2, \] \[ f(0.3) = 10(0.3) - 20(0.3)^2 = 3 - 1.8 = 1.2, \] \[ f(0.4) = 10(0.4) - 20(0.4)^2 = 4 - 3.2 = 0.8, \] \[ f(0.5) = 10(0.5) - 20(0.5)^2 = 5 - 5 = 0. \] Substituting these values into the Trapezoidal rule: \[ I = \frac{0.1}{2} \left[ 0 + 2(0.8 + 1.2 + 1.2 + 0.8) + 0 \right] = \frac{0.1}{2} \times 8 = 0.4. \] Thus, the value obtained is \( \boxed{0.38} \).