First slide

Area of bounded Regions

Question

$The area bounded by y = \frac{1}{x^{2} - 2 x + 2} and x -axis.$

Moderate

A

1
B

2
C

3
D

4

Solution

$y = \frac{1}{(x - 1)^{2} + 1}$
$When x = 1, y_{max} = 1 When x \to \pm \infty, y \to 0 Therefore, x -axis is the asymptote. Also f (1 + x) = f (1 - x)$
$Hence, the graph is symmetrical about line x = 1$
$From these information the graph of function is as shown in$ $adjacent figure.$
$Area = 2 \int_{1}^{\infty} \frac{1}{(x - 1)^{2} + 1} d x = {[2 \tan^{- 1} (x - 1)]}_{1}^{\infty} = π sq. units$

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