Question

Find the area of the region bounded by the curve $y=1+\frac{8}{x^2}$, x-axis and the equation x = 2 and x = 4.

Answer 1

Given the curve is $y=1+\frac{8}{x^2}$

So

(1). Curve is symmetric about Y-axis.

(2). As $x\to 0,y\to \infty$

(3). As $x\to \pm \infty ,y\to 1$

(4). Curve

$y-1=\frac{8}{x^2}>0$

$\Rightarrow y>1$

Curve exists only

Required Area

$={\int }_{x=2}^{4}ydx$

$={\int }_2^4(1+\frac{8}{x^2})dx$

$={{[x-\frac{8}{x}]}_{x=2}}^4$

$=(4-\frac{8}{4})-(2-\frac{8}{2})$

$=(4-2)-(2-4)$

$=2+2=4$

Hence, this is the answer.