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Math Help - Find the area of region between curves

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    Find the area of region between curves

    Sketch the region enclosed by the given curves.y = 2/x, y = 8x, y =1/8x x > 0


    Find its area.


    I found that this is the correct graph
    In order to find its area, do we need to take the integral of each side and add them together? How do we find the limits? Do we need to find in terms of x instead of the y= values they give us?

    I thought for graph problems you subtract the two equations but I'm not sure..
    Last edited by Steelers72; April 9th 2013 at 08:51 AM.
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    Forum Admin topsquark's Avatar
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    Re: Find the area of region between curves

    Quote Originally Posted by Steelers72 View Post
    Sketch the region enclosed by the given curves.y = 2/x, y = 8x, y =1/8x x > 0


    Find its area.


    I found that this is the correct graph
    In order to find its area, do we need to take the integral of each side and add them together? How do we find the limits? Do we need to find in terms of x instead of the y= values they give us?

    I thought for graph problems you subtract the two equations but I'm not sure..
    Start by finding the two intersection points. Then I'd integrate over x in the two regions.

    -Dan
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    Junior Member Mathhead200's Avatar
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    Re: Find the area of region between curves

    You do subtract then integrate; but as topsquark pointed out, you need to find the intersection point of the "curves" (i.e. functions) bounding the region on the top. (See your vertical blue line.) Then you can split the region up into two areas, bounded above and below by only two functions, (one area to the left of that blue line, and one to the right.)
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    Re: Find the area of region between curves

    More important than "thinking" that "for graph problem you subtract", you should understand why you subtract. Think back to "Riemann sums"- you want to break the area into many thin rectangles. The area of each rectangle is the "thickness", dx, times the length- that is the difference between the two formulas.
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    Re: Find the area of region between curves

    Thanks guys.

    Would the limits be .5 and 4? I followed the blue lines down and x= .5 and x=4

    so integral 0 to .5 [1/8x] - integral .5 to 4 [2/x] ? Am I on the right track?

    If so,
    Last edited by Steelers72; April 9th 2013 at 04:53 PM.
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    Junior Member Mathhead200's Avatar
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    Re: Find the area of region between curves

    Yes, but you should make sure x = .5 is the intersection point... with algebra...!
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    Re: Find the area of region between curves

    Let us first name the equations:
    y = 8x ---- 1.
    y = 2/x ---2.
    y = (1/8) x
    I would suggest that first you find the x coordinates of points of intersection.
    Intersection of 1 and 2 x = 1/2
    Intersection of 2 and 3 x = 4
    Now the required area = (area under the curve 1 from x = 0 to x = 1/2) + (area under the curve 2 from x = 01/2 to x = 4) - (area under the curve 3 from x = 0 to x = 4)
    Now use integration to get the result
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