For positive real numbers $R$ and $r$, let $$E(R, r) = \{\frac{x_1^2 + x_2^2 + x_3^2}{R^2} + \frac{x_4^2}{r^2}\leq 1\}$$ Using an iterated integral, calculate the volume of $E(R,r)$.
I am not sure if I did this right since I did not use an iterated integral: \begin{align} \int_{-r}^{r}R^2(1 - \frac{x_4^2}{r^2})\;dx_4 &= R^2(x_4 - \frac{x_4^3}{3r^2})\Big|^r_{-r} \\ &=2R^2(r - r / 3) \\ &= \frac{4R^2r}{3} \end{align}
With the change of coordinates suggested in the comment $E(R,r)$ is transformed into the unit ball $$D:y_1^2+y_2^2+y_3^2+y_4^2\leq 1$$ in $\mathbb R^4$. According to the rule for changing coordinates in multiple integrals, $$ \text{vol}\,(E(R,r))=\int_{E(R,r)}1\,dx_1\,dx_2\,dx_3\,dx_4= \int_D \frac{d\mathbf x}{d\mathbf y}\,dy_1\,dy_2\,dy_3\,dy_4 $$ where $$ \frac{d\mathbf x}{d\mathbf y} $$ is the Jacobian matrix determinant, i.e. the determinant of the matrix consisting of the partial derivatives $\frac{\partial x_j}{\partial y_k}$ in place $(j,k)$. In this case, the Jacobian matrix is diagonal, with diagonal entries $R$, $R$, $R$ and $r$. Thus $$ \frac{d\mathbf x}{d\mathbf y}=R^3r, $$ and $$ \text{vol}\,(E(R,r))=\int_D R^3r\,dy_1\,dy_2\,dy_3\,dy_4=R^3r\,\text{vol}\,(D). $$ Now, you are back on safe ground, if I understand it correctly.