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Homework Help: Area Between 2 Curves

  1. Dec 5, 2017 #1
    1. The problem statement, all variables and given/known data
    Calculate the area between ##y = -x^2+3x+10## and ##y = -x+14##. Note that ##y = -x+14## is the tangent to the curve ##y = -x^2+3x+10## at the point ##(2,12)##.

    2. Relevant equations


    3. The attempt at a solution
    Is it as simple as calculating ##\int_{5}^{14} -x+14+x^2-3x-10 = \int_{5}^{14} x^2-4x-10##?
     
  2. jcsd
  3. Dec 5, 2017 #2

    Mark44

    Staff: Mentor

    Where did the 5 come from?

    Anyway, your problem is not well-posed, as posted here. Is there some additional information that you left out? The area of the region between the line and the parabola is infinitely large. Did you sketch a graph?
     
  4. Dec 5, 2017 #3

    StoneTemplePython

    User Avatar
    Science Advisor
    Gold Member

    it depends on what you want out of this problem.

    your problem statement didn't say that the area is to be calculated over ##[5, 14]##, and you are missing a dx in those integrals, but otherwise it looks basically ok.

    two concerns:

    the problem statement asks for

    ##\int_{a}^{b} \big \vert (-x+14) - (-x^2+3x+10) \big \vert dx##

    how do you know that

    ##\big \vert (-x+14) - (-x^2+3x+10) \big \vert = (-x+14) - (-x^2+3x+10) ##

    ?

    Put differently, how do you know you can just get rid of those absolute value signs? In general this would require a justification. I would always start by graphing it / drawing a picture, but some symbolic / mathematical justification is still needed.

    you may also want to confirm ##14 - 10 \neq -10##
     
  5. Dec 6, 2017 #4
    Sorry about the confusion.
    The area is between the line and the parabola and above the x axis. My bounds were ##5 \leq x \leq 14## because that's where the line and curve cut the x axis after drawing it.
     
  6. Dec 6, 2017 #5

    Mark44

    Staff: Mentor

    That's still an infinite area, unless there are some other bounds. The region between the parabola and the line, and above the x-axis, is in two parts -- the roughly triangular piece to the right of the point where the two figures intersect, and the part to the left of the intersection point. The portion on the left is infinite in area.
     
  7. Dec 6, 2017 #6
     

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  8. Dec 6, 2017 #7

    StoneTemplePython

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    Science Advisor
    Gold Member

    your problem statement still is underspecified. Where did the integration bounds of [5,14] come from? For instance, why not also integrate from ##(-\infty, -2)## and why not [-2,2]?
     
  9. Dec 6, 2017 #8
    Thanks for the help.
    I just calculated ##\int_{2}^{15} -x+14 \; dx## and ##\int_{2}^{5} -x^2+3x-10 \; dx## then took the difference to get ##\frac{329}{6}##.
     
  10. Dec 6, 2017 #9

    Mark44

    Staff: Mentor

    There is also the area to the left of the parabola and below the line.
     
  11. Dec 6, 2017 #10
    Not according to the graph above.
     
  12. Dec 6, 2017 #11

    Mark44

    Staff: Mentor

    But the graph doesn't agree with your problem statement in post #1.
    The area between the line and the parabola includes the portion to the left of the parabola.
    The graph implies additional restrictions that aren't given in the problem statement -- i.e., that ##x \ge 2## and ##y \ge 0##.

    So which one is the problem you're working? The verbal description or the graph? They aren't the same.
     
  13. Dec 6, 2017 #12

    Mark44

    Staff: Mentor

    The 15 in the first integral will give you a wrong answer. Was that a typo?
     
  14. Dec 7, 2017 #13
    Sorry guys.
    The question was to find the shared area in the graph & yes that was a typo.
    Cheers.
     
  15. Dec 7, 2017 #14

    Mark44

    Staff: Mentor

    Do you mean "shaded" area?
     
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