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Fundamental Theorem of Calculus

  1. Aug 27, 2012 #1
    1. The problem statement, all variables and given/known data

    All this information is in the attached file.

    2. Relevant equations

    All this information is in the attached file.


    3. The attempt at a solution


    What I tried to do was take the anti-derivative of the first equation and plug in the number 5. I'm not sure if that was the way to go so I then just plugged in 5 off the get-go and got 13. Not sure what the question is asking and I would appreciate some guidance.

    Thanks
     

    Attached Files:

  2. jcsd
  3. Aug 27, 2012 #2

    Mark44

    Staff: Mentor

    If you have concluded that F(5) = 13, then no, that is incorrect. For that part you should NOT attempt to get the antiderivative.
     
  4. Aug 27, 2012 #3
    So the anti-derivative is only applicable to parts b and c?
     
  5. Aug 27, 2012 #4

    Mark44

    Staff: Mentor

    You should not attempt to get the antiderivative in any of the three parts. For part b, you'll need to use the Fundamental Theorem of Calculus. Note that there are two parts of this theorem.
     
  6. Aug 27, 2012 #5
    The book says to use the second part of the theorem but I'm not seeing how it would work in this question. Do I plug in the x in place of t?
     
  7. Aug 27, 2012 #6

    Mark44

    Staff: Mentor

    You're talking about part a, right? What definite integral is represented by F(5)?
     
  8. Aug 27, 2012 #7
    √5^2 + 144?
     
  9. Aug 27, 2012 #8

    Mark44

    Staff: Mentor

    No.
    Show me what the definite integral looks like. If you quote this post you should be able to reuse this integral.

    $$F(5) = \int_?^? \sqrt{t^2 + 144}~dt $$
     
  10. Aug 27, 2012 #9
    Is it that?
     
  11. Aug 27, 2012 #10

    Mark44

    Staff: Mentor

    Yes. Now, what is the value of that integral? That is what F(5) equals. Don't overthink this.
     
  12. Aug 27, 2012 #11

    SammyS

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    ralfsk8,

    In general, what does the definite integral,
    [itex]\displaystyle \int_{a}^{b}\, f(x)\,dx[/itex]​
    represent?
     
  13. Aug 27, 2012 #12
    Mark 44 - I would say either 13 or 0, but as you mentioned earlier, 13 isn't the answer.

    SammyS - The area under the curve bounded by the x-axis of a graph and 2 vertical lines (a and b)
     
  14. Aug 27, 2012 #13

    Mark44

    Staff: Mentor

    What are the vertical lines for this integral?
     
  15. Aug 27, 2012 #14
    The vertical lines for this integral are 5 and x
     
  16. Aug 27, 2012 #15

    Mark44

    Staff: Mentor

    That's for F(x). You're trying to determine F(5). So what are the vertical lines now?
     
  17. Aug 27, 2012 #16

    Zondrina

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    An easy way to remember the fundamental theorem is to consider :

    Let :

    [itex]F(x) = \displaystyle \int_{b(x)}^{a(x)}\, f(t)\,dt[/itex]

    Then :

    [itex]\frac{d}{dx}F(x) = \frac{d}{dx} \displaystyle \int_{b(x)}^{a(x)}\, f(t)\,dt = f(a(x))a'(x) - f(b(x))b'(x)[/itex]

    It's simply a plug and play formula.
     
  18. Aug 27, 2012 #17
    Mark44 - The vertical lines are now 5 and 5. So we're not trying to calculate the area of anything if it's the same line?

    Zondrina - Ah yes, that does seem familiar to me
     
  19. Aug 27, 2012 #18
    Mark44 - I would assume 5 to 5 which would make it the same line right?

    Zondrina - Ah yes, that does seem familiar
     
  20. Aug 27, 2012 #19

    Zondrina

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    Also think about this F(5) thing, it's MUCH easier than it looks. Imagine you're integrating from one line a to another line b. What if those two lines are the same? Then there really is no area between them right?

    A simple example is to consider :

    [itex]\int_{z}^{z}xdx[/itex]

    Where z is ANY real number or function or anything really, but to keep it simple let z = 1,2,3...... and notice no matter what you're going to get the same answer. Then come back to your F(5) question.
     
  21. Aug 27, 2012 #20

    Mark44

    Staff: Mentor

    Right. So F(5) = ?
     
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