Find Domain of F(x): Homework Statement

[theJorge551]

Homework Statement

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

Where {0}\leq{t}\leq{4}.

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?

 

[Tedjn]

Why is there a condition 0 ≤ t ≤ 4? The variable t is only used for the integration and has no bounds.

 

[slider142]

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.

 

[theJorge551]

Ahh, thank you for the clarafication, slider. I had a suspicion that the domain of x is the same as the domain of t.

 

[hgfalling]

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.

 

[theJorge551]

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?

 

[hgfalling]

Well, suppose you want to evaluate F(3). Then you need
\intop_1^9 \frac{10}{2+t^3} dt

But t is supposed to be between 0 and 4, right? So this expression is undefined.

 

[theJorge551]

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].

 

[hgfalling]

I agree.