Trying to find the area between two curves

  • Thread starter Triathlete
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  • #1
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A circle with radius 1 touches the curve y = |2x| in two places(see attachment for picture). Find the area of the region that lies between the curves.

I am having a tough time with this one. I figured I could put the radius in a spot where it would form a right angle on the line, then try and split the area into two large rectangles, and then subtract the corresponding sectors of the circle from the volume.
Only problem is, there aren't really any numbers so even though I know how to use integration to find the arc length for the sectors, I don't know WHAT to integrate or what my limits of integration would be. I can't seem to find a good place to start. Any help would be appreciated.
 

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Answers and Replies

  • #2
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I have attached another picture that may better explain my 'game plan'.
 

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  • #3
chiro
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Hey Triathlete.

There are (like all problems in mathematics) many ways to do the problem, but one suggestion is to find the equation of the circle and find the difference of the triangle and that of the cone.

Note that if you integrate with respect to the x-axis then what you will get is the area under the circle which will look like a skate-ramp with a block under-neath it.

You can use the idea that the area of the triangle-part can be decomposed from the rectangular part to get the area of the region outside of the triangle.

When you have the area under the circle curve along the x-axis and the region outside the triangle, you take (1) - (2) and you will get the area of the triangle.

So your region of integration will be where the circle intersects the triangle. Then you need to get the equation for your circle so you can find the area under that and the rest relies on splitting up the area in that region to get the triangle separated from the whole rectangle.
 

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