Find area of shaded region
[img=http://img411.imageshack.us/img411/9/areamh2.th.jpg]
Find area of shaded region
[img=http://img411.imageshack.us/img411/9/areamh2.th.jpg]
Well, we know that:
$\displaystyle \left|\int_a^b [f(x)]dx\right| = Area$
Well, we know the following:
$\displaystyle a = 1$ We know this because the -1 at the end shifts it to the right 1.
$\displaystyle b = 4$ It is labeled as the end of the region.
$\displaystyle y = f(x) = \frac{1}{x^2} - 1$
So, therefore:
$\displaystyle \left|\int_1^4 [\frac{1}{x^2} - 1]dx\right|$
So now we have to rewrite it in a way that would make this easier to integrate:
$\displaystyle \left|\int_1^4 [x^{-2} - 1]dx\right|$
Now we integrate:
$\displaystyle |[-x^{-1} - x]_1^4|$
$\displaystyle \left|\left[\left(-\frac{1}{4} - 4\right) - (-1 - 1)\right]\right|$
$\displaystyle \left|\left[\left(-\frac{5}{4}\right) + 2\right]\right|$
$\displaystyle \left|\frac{3}{4}\right| \text{units}^2 = \frac{3}{4} \text{units}^2 = Area$
There you go.