Question

Find the mass of the solid in Problem 48 if the density is z.

Short answer

The required Mass of a solid is $\text{182.04}$.

Step-by-step solution

Definition of Mass of a solid

A solid mass is the amount of a substance contained by a body, measured in Kilograms.

Finding the Mass of a solid

Consider the given density z,

Now, find the mass:

$$\begin{gathered} \text{M =}{\int }_{\text{0}}^{\text{4}}\text{dx}{\int }_{\text{0}}^{\text{x}}\text{dy}{\int }_{\text{0}}^{{\text{x}}^{\text{2}}{\text{-y}}^{\text{2}}}\text{zdz} \\ \text{=}{\left.{\int }_{\text{0}}^{\text{4}}\text{dx}{\int }_{\text{0}}^{\text{x}}\text{dy}\frac{\text{1}}{\text{2}}{\text{z}}^{\text{2}}\right|}_{\text{0}}^{\text{2}}{\text{-y}}^{\text{2}} \\ \text{=}\frac{\text{1}}{\text{2}}{\int }_{\text{0}}^{\text{4}}\text{dx}{\int }_{\text{0}}^{\text{x}}\text{dy}\left({\text{x}}^{\text{4}}{\text{- 2x}}^{\text{2}}{\text{y}}^{\text{2}}{\text{+y}}^{\text{4}}\right) \\ \text{=}{\left.\frac{\text{1}}{\text{2}}{\int }_{\text{0}}^{\text{4}}\text{dx}\left({\text{x}}^{\text{4}}\text{y -}\frac{\text{2}}{\text{3}}{\text{x}}^{\text{2}}{\text{y}}^{\text{3}}\text{+}\frac{{\text{y}}^{\text{5}}}{\text{5}}\right)\right|}_{\text{0}}^{\text{x}} \end{gathered}$$

The final value is obtained using further calculation –

$$\begin{gathered} \text{M =}\frac{\text{1}}{\text{2}}{\int }_{\text{0}}^{\text{4}}\left({\text{x}}^{\text{5}}\text{-}\frac{\text{2}}{\text{3}}{\text{x}}^{\text{5}}\text{+}\frac{{\text{x}}^{\text{5}}}{\text{5}}\right)\text{dx} \\ \text{=}\frac{\text{4}}{\text{15}}{\int }_{\text{0}}^{\text{4}}{\text{x}}^{\text{5}}\text{dx} \\ \text{=}\frac{\text{4}}{\text{90}}{\text{x}}^{\text{6}}{\int }_{\text{0}}^{\text{4}} \end{gathered}$$

$$\begin{gathered} \text{M =}\frac{\text{8192}}{\text{45}} \\ \text{= 182.04} \end{gathered}$$

Hence, the Mass of a solid is$182.04$ .