Question

Verify that (3.10) gives the same result as (3.8).

Short answer

The given equation is verified.

Step-by-step solution

Representation of the equations

Write equation 3.10.

$\mathit{V}\mathbf{=}{\mathbf{\int }}_{\mathbf{x}\mathbf{=}\mathbf{0}}^{\mathbf{1}}{\mathbf{\int }}_{{}_{\mathbf{(}}\mathbf{z}\mathbf{=}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}^{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{\int }}_{\mathbf{y}\mathbf{-}\sqrt{\mathbf{(}{\mathbf{x}}^{\mathbf{4}}\mathbf{-}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}}}^{\sqrt{\mathbf{(}{\mathbf{x}}^{\mathbf{4}}\mathbf{-}{\mathbf{z}}^{\mathbf{2}}\mathbf{)}}}\mathbf{=}\mathit{d}\mathit{y}\mathit{d}\mathit{z}\mathit{d}\mathit{x}$

Write equation 3.8.

$$\begin{gathered} \mathit{V}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\mathit{\pi }{\mathit{y}}^{\mathbf{2}}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{1}}\mathit{\pi }{\mathit{x}}^{\mathbf{4}}\mathit{d}\mathit{x} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{5}} \end{gathered}$$

Verification of the given equation

Simplify equation 3.10.

$$\begin{gathered} V={\int }_0^{1}dx{\int }_{-x^2}^{x^2}dz{\int }_{-\sqrt{x^4-z^2}}^{\sqrt{x^4-z^2}}dy \\ ={\int }_0^{1}dx{\int }_{-x^2}^{x^2}2\sqrt{x^4-z^2} \\ =2{\int }_0^{1}dx{\int }_{-x^2}^{x^2}\sqrt{x^4-z^2}dz \end{gathered}$$

It is known that $z=x^{2}sinu,anddz=x^{2}cosudu$Substitute the values in the above expression.