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I am given $y=x$ and $y=\frac{x}{8}$ for the lines. The curve is $y=\frac{1}{x^2}$ . This is more of a question about the question though. I have no idea what it is asking.

My instinct is to integrate over some limits but I am not sure where this circle features and whether I am over complicating things.

Should I just be adding 3 indefinite integrals here?

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    that the curve was a circle. I don't know why I did that I have it as an asymptote!2012-08-12

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Make a graph showing all three curves. You’ll see that there is just one finite region bounded by these curves. It has two straight sides along the lines $y=x$ and $y=x/8$ and one curved side along $y=1/x^2$. If that third side ran straight from $\langle 1,1\rangle$ to $\left\langle 2,\frac14\right\rangle$, the region would be the triangle with vertices at the origin, at $\langle 1,1\rangle$, and at $\left\langle 2,\frac14\right\rangle$.

(In case it isn’t clear, $\langle 1,1\rangle$ is the point of intersection of $y=x$ and $y=1/x^2$, and $\left\langle 2,\frac14\right\rangle$ is the point of intersection of $y=x/8$ and $y=1/x^2$.)

You’ll want to set it up as two integrals, one from $x=0$ to $x=1$, the other from $x=1$ to $x=2$. Your vertical strips $dA$ of area for the first integral will run between the two straight lines; for the second integral they’ll run from the line $y=x/8$ to the curve $y=1/x^2$.

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    @Magpie: Yes, that looks fine.2012-08-12