Q29P In Problems 17 to 30, for the cu... [FREE SOLUTION]

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 $x 鈮� 0$ octants is equal. So,it can be integrated over the $y 鈮� 0$ , $z 鈮� 0$ octant and just multiply the result by 4.

Determine the mass of the volume by Integrating over z .

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

Determination of the mass of the solid of revolution

Integrate over x and y .

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

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

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91影视

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

Fundamentals of Physics

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Study anywhere. Anytime. Across all devices.

91影视

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

Fundamentals of Physics

Nuclear Physics

Electromagnetism

Solid State Physics

Measurements

Geometrical and Physical Optics

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 $| xyz |$ .