Chapter 5: Q29P (page 257)

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between x=0and x=2, find:
The mass of the solid of revolution if the density (mass per unit volume) is $| xyz |$ .

Short Answer

Expert verified

The mass of the solid of revolution if the density is2.

Step by step solution

Definition of solid of revolution

The solid that is obtained on rotating a figure which is plane in nature around a particular axis of the straight line is known as solid of revolution.

Representation of the area bounded by the curve

Draw the bounded region for the curve.

Determination of the mass

The mass contained in each of the $x \geq 0$ octants is equal. So,it can be integrated over the $y \geq 0$ , $z \geq 0$ octant and just multiply the result by 4.

Determine the mass of the volume by Integrating over z .

$M = 4 \int_{0}^{2} x d x [\int_{0}^{f (x)} y d y \int_{0}^{\sqrt{f^{2} (x) - y^{2}}} z d z] = 4 \int_{0}^{2} x d x [\int_{0}^{\sqrt{x}} y d y \int_{0}^{\sqrt{x - y^{2}}} z d z] = 4 \int_{0}^{2} x d x [\int_{0}^{\sqrt{x}} y d y {\frac{1}{2} z^{2}}_{0}^{\sqrt{x - y^{2}}}] = 2 \int_{0}^{2} x d x [\int_{0}^{\sqrt{x}} y d y (x - y^{2})]$

Determination of the mass of the solid of revolution

Integrate over x and y .

$M = 2 \int_{0}^{2} x d x {[\frac{1}{2} x y^{2} - \frac{1}{4} y^{4}]}_{0}^{\sqrt{x}} = 2 \int_{0}^{2} x d x \frac{x^{2}}{4} = 2 \int_{0}^{2} \frac{x^{3}}{2} d x = {\frac{1}{8} x^{4}}_{0}^{2} = 2$

Thus, mass of the solid of revolution if the density is 2.

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In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between x=0and x=2, find:
The mass of the solid of revolution if the density (mass per unit volume) is $| xyz |$ .

Short Answer

Step by step solution

Definition of solid of revolution

Representation of the area bounded by the curve

Determination of the mass

Determination of the mass of the solid of revolution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

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Study anywhere. Anytime. Across all devices.

In Problems 17 to 30, for the curve y=x, between x=0and x=2, find:The mass of the solid of revolution if the density (mass per unit volume) is |xyz|.

Short Answer

Step by step solution

Definition of solid of revolution

Representation of the area bounded by the curve

Determination of the mass

Determination of the mass of the solid of revolution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Energy Physics

Physical Quantities And Units

Thermodynamics

Mechanics and Materials

Measurements

Study anywhere. Anytime. Across all devices.

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between x=0and x=2, find:
The mass of the solid of revolution if the density (mass per unit volume) is $| xyz |$ .