Induced Riemannian volume of an immersed submanifold

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Problem 3.3.2 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let $C = \{ x \in \mathbb {R}^{3} : 0 \leq x_i \leq 1\}$ be the unit cube in $\mathbb {R}^{3}$ . Suppose $F : \mathbb {R}^{3} \to \mathbb {R}^{4}$ is a $1-1$ $C^\infty$ immersion in some neighborhood of $C$ . The image $F(C)$ is then a compact Riemannian submanifold of $\mathbb {R}^{4}$ with boundary and therefore has a volume. Justify the following formula:

\operatorname {vol}(F(C)) = \int _0^1\int _0^1\int _0^1 \left\| F_*\left(\frac{\partial }{\partial x_1}\right) \wedge F_*\left(\frac{\partial }{\partial x_2}\right) \wedge F_*\left(\frac{\partial }{\partial x_3}\right)\right\| \, dx_1 \, dx_2 \, dx_3.

Also, evaluate the integrand if

F(x) = ((1+x_1)^2, (1+x_2)^2, (1+x_3)^2, (2+x_3)^3).

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.