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Area of bounded Regions

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By Expert Faculty of Sri Chaitanya

Question

$The area of the region \{(x, y) : 0 \leq y \leq x^{2} + 1, 0 \leq y \leq x + 1, 0 \leq x \leq 2\} is$

Moderate

Solution

Equations of given curves are
$y = x^{2} + 1 \dots \dots \dots \dots \dots \dots$ (1)
$y = x + 1 \dots \dots \dots \dots \dots \dots \dots . (2)$
Solving (1) & (2)
$x^{2} + 1 = x + 1 \Rightarrow x^{2} - x = 0$
$x = 0, x = 1$
$y = 1, y = 2$
$(1) \underline{\underline{&} (2)} intersect at A (0, 1) and B (1, 2)$
$R A = \int_{0}^{1} (x^{2} + 1) d x + \int_{1}^{2} (x + 1) d x$
$= {(\frac{x^{3}}{3} + x)}_{0}^{1} + {(\frac{x^{2}}{2} + x)}_{1}^{2} = \frac{23}{6} = 3.83$

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