I would like to evaluate $\int_G dx^1\wedge \cdots \wedge dx^k$ where $G \subset \mathbb{R}^{k+1}$ is the graph of the function $F : [0,1]^k\to \mathbb{R}$.
Intuitively, I think this is the "volume" of $F$ over $[0,1]^k$. But when I expanded it out, my calculation showed it is something else. If I write $G(x^1, \cdots, x^k) = (x^1, \cdots, x^k, f(x^1,\cdots,x^k))$, then $$ \begin{aligned} \int_G dx^1\wedge \cdots \wedge dx^k &= \int_{[0,1]^k} G^*(dx^1\wedge \cdots \wedge dx^k) \\ &= \int_{[0,1]^k} (G^*dx^1) \wedge \cdots \wedge(G^* dx^k) \\ &= \int_{[0,1]^k} dx^1\wedge \cdots \wedge dx^k = 1^k = 1 \end{aligned} $$ Somehow $f$ does not come into play here. Can someone point out where I did something wrong? Thanks in advance!