Find area of shaded region

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- Mar 10th 2008, 10:33 AMgraceyshaded region
Find area of shaded region

[img=http://img411.imageshack.us/img411/9/areamh2.th.jpg] - Mar 10th 2008, 01:22 PMAryth
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.