Referring to the shaded bounded region $R$ shown in the accompanying figure, bounded between the parabola $y = x^2$ and the straight line $y = 2x$ intersecting at $(0,0)$ and $(2,4)$, the area of region $R$ evaluated via double integration $\iint_R dx \, dy$ is ________ (round off to two decimal places).
Answer Key:
1.33
Step-by-Step Solution & Explanation
1. Points of intersection:
$x^2 = 2x \implies x(x - 2) = 0 \implies x = 0, x = 2$
2. In region $R$, for any $x \in [0, 2]$, the upper curve is $y = 2x$ and lower curve is $y = x^2$.
3. Area via double integral:
$A = \int_{0}^{2} \int_{x^2}^{2x} dy \, dx = \int_{0}^{2} (2x - x^2) \, dx = \left[ x^2 - \frac{x^3}{3} \right]_0^2 = 4 - \frac{8}{3} = \frac{4}{3} \approx 1.33$.