I have three equations: $y=3/x$, $y=12x$, and $y=x/12$, $x>0$. I am not sure how to go about integrating an equation once I find the intersections. Do I need multiple integrals?

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The region within the three curves can be plotted and its area determined using Mathematica's geometric capabilities.

RegionPlot[y < 3/x && y < 12 x && y > x/12, {x, 0, 6}, {y, 0, 6}, 
    PlotPoints -> 200, FrameLabel -> {x, y}]

enter image description here

and integrate its area by

Area[ImplicitRegion[y < 3/x && y < 12 x && y > x/12, {x, y}]]

(* Log[1728] *)
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When you use Mathematica, you could start with a plot like this

Plot[{3/x, 12 x, x/12}, {x, 0, 10}, PlotRange -> {{0, 10}, {0, 10}}]

Then you can see where the area is which you should integrate.

And by using Solve[3/x == 12 x, x], Solve[12 x == x/12, x] and Solve[x/12 == 3/x, x] you can calculate the three intersections which are 0, 0.5 and 6.

The area between the curves can then be calculated by

Integrate[12 x, {x, 0, 0.5}] + Integrate[3/x, {x, 0.5, 6}] - Integrate[x/12, {x, 0, 6}].

I hope this help.

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