# Find domain of F(x)?

## 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.

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?

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

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.

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

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.

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?

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.

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].

I agree.