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Chapter 5: Q50P (page 249)

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

Short Answer

Expert verified

The required Mass of a solid is $182.04$ .

Step by step solution

01

Definition of Mass of a solid

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

02

Finding the Mass of a solid

Consider the given density z,

Now, find the mass:

$M = \int_{0}^{4} dx \int_{0}^{x} dy \int_{0}^{x^{2} {-y}^{2}} zdz = {\int_{0}^{4} dx \int_{0}^{x} dy \frac{1}{2} z^{2}|}_{0}^{2} {-y}^{2} = \frac{1}{2} \int_{0}^{4} dx \int_{0}^{x} dy (x^{4} {- 2x}^{2} y^{2} {+y}^{4}) = {\frac{1}{2} \int_{0}^{4} dx (x^{4} y - \frac{2}{3} x^{2} y^{3} + \frac{y^{5}}{5})|}_{0}^{x}$

The final value is obtained using further calculation –

$M = \frac{1}{2} \int_{0}^{4} (x^{5} - \frac{2}{3} x^{5} + \frac{x^{5}}{5}) dx = \frac{4}{15} \int_{0}^{4} x^{5} dx = \frac{4}{90} x^{6} \int_{0}^{4}$

$M = \frac{8192}{45} = 182.04$

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

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