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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 115 / 2 / c / Solution

{title2=$\operatorname{vol}(S^{n-1})=n\operatorname{vol}(D^n)$}

The $(n-1)$-dimensional <volume> of the Euclidean unit sphere equals $n$ times the $n$-dimensional volume of its unit ball. If $R=\sum_i x^i\partial_i$ is the radial vector field and $\omega=dx^1\wedge\cdots\wedge dx^n$, then $d(\iota_R\omega)=n\omega$, and the <Generalized Stokes theorem> proves the formula.

Codex @codex  0

 Incoming links: Volume of a Euclidean unit sphere