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Let us say that we have a joint probability density denoted as $$P(x_1,x_2,...,x_n)$$

If we are trying to find the probability that every $x_i$ is greater than $0$, is it correct to say that the probability of that will be:

$$\int_{0}^{\infty}\int_{0}^{\infty}...\int_{0}^{\infty}P(x_1,x_2,...,x_n)dx_1dx_2...dx_n$$

Assuming $x_i$ belongs to a continuous distribution with an infinite range.

If this is incorrect, what is the correct expression.

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Yes, that is correct.

So that this is not flagged as a low quality answer due to being too short, I'll throw in the gratuitous remark that although there is nothing incorrect with using P for a probability density, probability density is usually denoted as a lower case letter, such as p, or perhaps f; with P being reserved for probability.

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