Find domain of F(x)?

  • #1

Homework Statement



[tex]F(x) \ = \ \int\limits_{1}^{x^2}{\frac{10}{2+t^3}} \ dt[/tex]

Where [tex]{0}\leq{t}\leq{4}[/tex].

Find the domain of F.

Homework Equations



N/A

The Attempt at a Solution



I'm not quite sure how to tackle this problem. It doesn't seem as though the domain of t has much at all to do with the domain of F(x), so could anyone steer me in the correct direction on how to approach this?
 

Answers and Replies

  • #2
737
0
Why is there a condition 0 ≤ t ≤ 4? The variable t is only used for the integration and has no bounds.
 
  • #3
1,013
70
I believe the limits on t define the domain of the function being integrated. That is, the integral is not defined if the limits of the integral venture outside the domain of the internal function.
 
  • #4
Ahh, thank you for the clarafication, slider. I had a suspicion that the domain of x is the same as the domain of t.
 
  • #5
351
1
The acceptable values for t are not the same as the domain of F (ie, the acceptable values for x). I can't tell from the last comment whether you were saying that you thought that or not.
 
  • #6
Because F(x) is independent of the values of t, I think that the domain of F is the set of real numbers. The value of t being constricted to the area between 0 and 4 should have nothing to do with the constriction of x as it varies, correct?
 
  • #7
351
1
Well, suppose you want to evaluate F(3). Then you need
[tex] \intop_1^9 \frac{10}{2+t^3} dt [/tex]

But t is supposed to be between 0 and 4, right? So this expression is undefined.
 
  • #8
All right, so on the upper end of x, the limit is 2 (2^2 = 4), and it can go down to -2 before becoming undefined yet again...so the domain of F would be [-2,2].
 
  • #9
351
1
I agree.
 

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