I'm studying for a term test coming up and I'm practicing by doing some change of variables questions. There's one I'm stuck on, and it says "Let $A$ be a Jordan-measurable subset of $\mathbb{R}^{n-1}$. Let $p=(p_1,\dots,p_n)\in\mathbb{R}^n$ with $p_n>0$, and let $C\in\mathbb{R}^n$ denote the cone with base $A\times\{0\}$ sand vertex $p$, defined as $$C=\{y\in\mathbb{R}^n:y=(1-t)x+tp,\ \text{where}\ x\in A\times\{0\},\ t\in[0,1]\}.$$ Show that $C$ is Jordan-measurable."
First of all, I know that the base of the cone is Jordan-measurable, since $A$ is Jordan-measurable (and therefore $A\times\{0\}$ is Jordan-measurable). However, I'm stuck on showing that the rest of the cone is Jordan-measurable, and I cannot see how this problem relates to a change of coordinates.
Help is greatly appreciated.
Using the definitions of the Riemann integral and the Jordan content, we have
$J(C)=\int_{R^n}1_C(y)dy$ whenever this integral exists. So it suffices to compute the integral.
Let $A_t=\{((1-t)x_1,\cdots,(1-t)x_{n-1})_t:(x_1,\cdots,x_{n-1})\in A\}$. Then, $J(A_t)=(1-t)^{n-1}J(A)\ $ (since $A_t$ is Lebesgue measurable for $0\le t\le 1).$
To finish, apply Fubini's theorem, using the fact that the points of $C$ are given by
$\{((1-t)(x_1+p_1),\cdots,(1-t)(x_{n-1}+p_{n-1}),tp_n):0\le t\le 1\}.$ That is,
$J(C)=\int_\mathbb{R}\left(\int_{\mathbb R^{n-1}}1_C(y)d\vec x\right)dy_n=p_nJ(A)\int^1_0(1-t)^{n-1}dt=\frac{p_nJ(A)}{n}$.