Find the volume of the shapes defined as: $n$-dimensional cone
Can this be done without using spherical coordinates?
$$i)\ \frac{x_1^2}{a_1^2}+\cdots+\frac{x_{n-1}^2}{a_{n-1}^2}=\frac{x_n^2}{a_n^2}, \quad 0\leq x_n \leq a_n.$$
$$ii)\ \text{The shape bounded by the following:} \\ \sqrt{\frac{x}{a}} + \sqrt{\frac{y}{b}}=1, \quad \sqrt{\frac{x}{a}}+\sqrt{\frac{y}{b}}=2, \\ \frac{x}{a} = \frac{y}{b}, \quad 4 \frac{x}{a} = \frac{y}{b} \  \ (a,b > 0)$$
Im thinking that some of these last equations are cones too? Im not sure. What geometric shapes are they to begin with? Does anyone have a link to a site, that explains these shapes?

It can be, but it's usually not because the equations are really messy. Almost no problem is only solvable in a given coordinate system, it's just a question of which integrals are less awful
