Find the area of the region enclosed by parabola $y = x^{2} + 2$ and lines $y = x, x = 3$ and $x = 0$

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Solution

$y = x^{2} + 2$ intersects Y-axis at $(0, 2)$
Area of a region bounded by a Curve
$y = x^{2} + 2$ , line $y = x, x = 3$ and $x = 0$ is $A = | I |$
where $I = \int_{0}^{3} (y_{1} - y_{2}) d x = \int_{0}^{3} (x^{2} + 2 - x) d x = {[\frac{x^{3}}{3} - \frac{x^{2}}{2} + 2 x]}_{0}^{3} = 9 + 6 - \frac{9}{2}$

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