= Solution
On the Euclidean <unit ball> $D^n$,
$$
\omega_X=dx^1\wedge\cdots\wedge dx^n.
$$
The outward <unit normal> along $S^{n-1}$ is the radial vector field $N=\sum_i x^i\partial_i$, so part b gives
$$
\omega_{\partial X}
=\sum_{i=1}^n(-1)^{i-1}x^i\,dx^1\wedge\cdots\wedge\widehat{dx^i}\wedge\cdots\wedge dx^n.
$$
Let $R=\sum_i x^i\partial_i$ on the ball and put $\beta=\iota_R\omega_X$. Direct use of the <exterior derivative> gives $d\beta=n\omega_X$. Therefore the <Generalized Stokes theorem> yields
$$
\int_{\partial X}\omega_{\partial X}
=\int_{\partial X}F^*\beta
=\int_Xd\beta
=n\int_X\omega_X.
$$
This proves the <volume of a Euclidean unit sphere> formula.
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