1. ## integration/signed area question

Find the area enclosed by the curve $\displaystyle f(x)= 1-1/x^2$, the x-axis, the line x=1/2 and x=2.

For signed areas, do I take the absolute value of sections with negative areas or do I add all of the areas up as they are? For this particular question I get a final answer of $\displaystyle A=0$ by adding the areas as they are but the answer in the textbook is $\displaystyle A=1$ (and this is found by taking the absolute of the negative area between [1/2, 1] ) I'd like to be clarified on which method to use ...From what I understand, signed areas do not take absolute values of negative areas. What am I doing wrong?

2. Originally Posted by shawli
Find the area enclosed by the curve $\displaystyle f(x)= 1-1/x^2$, the x-axis, the line x=1/2 and x=2.

For signed areas, do I take the absolute value of sections with negative areas or do I add all of the areas up as they are? For this particular question I get a final answer of $\displaystyle A=0$ by adding the areas as they are but the answer in the textbook is $\displaystyle A=1$ (and this is found by taking the absolute of the negative area between [1/2, 1] ) I'd like to be clarified on which method to use ...From what I understand, signed areas do not take absolute values of negative areas. What am I doing wrong?
$\displaystyle A = -\int_{\frac{1}{2}}^1 1 - \frac{1}{x^2} \, dx + \int_1^2 1 - \frac{1}{x^2} \, dx$

3. Find

$\displaystyle \int_1^2 1 - \frac{1}{x^2} dx+\left|\int_{\frac{1}{2}}^1 1 - \frac{1}{x^2} dx \right|$

4. Thank you.

What about if the boundaries were between the lines y=-1/2 and y=1/2?