Struggling to Calculate Integral of Differential Forms

Question

The problem states this: "Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function. Orient the graph $X = \Gamma_f$ of $f$ by requiring that the diffeomorphism $\phi: \mathbb{R}^n \to X\quad x \mapsto (x,f(x))$ be orientation preserving. Given a bounded open set $U$ in $\mathbb{R}^n$ compute the Riemannian volume of the image $X_U = \phi(U)$ of $U$ in $X$ as integral over $U$." In a previous problem I proved that $\phi^*\sigma = \left(1+\sum_{i=1}^n \left(\frac{\partial f}{\partial x_i}\right)^2\right)^{\frac{1}{2}}dx_1 \wedge \cdots \wedge dx_n$ where $\sigma$ is the volume form, so I know that can be substituted in. But from there I am confused on how to actually calculate the integral. My book doesn't really give any help on how to calculate it, but still asks this problem. This also comes before the proof of Stokes's. Any help would be greatly appreciated!