Question

Consider the equation for a curve, $y = f(x) = x^2 + x$. The area enclosed by the curve, the x-axis (y = 0 line); the vertical lines passing through x = 1 and x = 2 is ______(rounded off to 2 decimal places)

Answer 1

Correct Answer: 3.7

To find the area under the curve from \( x = 1 \) to \( x = 2 \), we integrate the function \( x^2 + x \): \[ A = \int_{1}^{2} (x^2 + x) \, dx \] Evaluating this integral, we compute: \[ A = \left( \frac{x^3}{3} + \frac{x^2}{2} \right]_{1}^{2} \] \[ A = \left( \frac{2^3}{3} + \frac{2^2}{2} \right) - \left( \frac{1^3}{3} + \frac{1^2}{2} \right) \] \[ A = \left( \frac{8}{3} + 2 \right) - \left( \frac{1}{3} + \frac{1}{2} \right) \] \[ A = \left( \frac{14}{3} \right) - \left( \frac{2}{3} \right) \] \[ A = \frac{12}{3} = 4 \, \text{square units} \]