Question

The area under the curve \( y = x^2 + 2x \) between \( x = 0 \) and \( x = 4 \), using the trapezoidal rule with a step size of one, (in integer) is .................

Hint:

- The trapezoidal rule is a numerical integration method that approximates the area under a curve by dividing the area into trapezoids.

Answer 1

The trapezoidal rule for approximating the area under a curve is given by: \[ A \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \] where \( h \) is the step size and \( x_0 \) and \( x_n \) are the endpoints. In this case, the function is \( y = x^2 + 2x \), the interval is from \( x = 0 \) to \( x = 4 \), and the step size is 1. The points of evaluation are \( x_0 = 0, x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4 \). First, calculate the function values at these points: \[ f(0) = 0^2 + 2(0) = 0, f(1) = 1^2 + 2(1) = 3, f(2) = 2^2 + 2(2) = 8, \] \[ f(3) = 3^2 + 2(3) = 15, f(4) = 4^2 + 2(4) = 24 \] Now, apply the trapezoidal rule: \[ A \approx \frac{1}{2} \left[ f(0) + 2(f(1) + f(2) + f(3)) + f(4) \right] \] \[ A \approx \frac{1}{2} \left[ 0 + 2(3 + 8 + 15) + 24 \right] \] \[ A \approx \frac{1}{2} \left[ 0 + 2(26) + 24 \right] = \frac{1}{2} \left[ 52 + 24 \right] = \frac{1}{2} \times 76 = 38 \] Final Answer: The area under the curve is \( \boxed{38} \).