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Chapter 6: Q. 26 (page 556)

The mass of the solid of revolution obtained by rotating the graph of y = 4.5 − 0.5 $x^{2}$ on [0, 3] around the y-axis and whose density at height y is ρ( y) = 1.3 − 0.233y ounces per cubic inch.

Short Answer

Expert verified

$6 π$

Step by step solution

Step1. Given Information

Equation of graph,

$y = 4.5 - 0.5 x^{2}$

Equation of density,

$ρ (y) = 1.3 - 0.233 y$

Step2. Calculate

$M = \int_{0}^{2 π} \int_{0}^{π} \int_{1}^{2} ρ \sin ϕ d ρ d ϕ d θ \iint \frac{3}{2} \sin ϕ d ϕ d θ = 3 \int d θ = 6 π$

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Most popular questions from this chapter

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$π \int_{1}^{5} {(\frac{y - 1}{2})}^{2} dy$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$

Solve the initial-value problem

\frac{d y}{d x} = 2 y (1 - 5 y), y (0) = 1

Each of the definite integrals in Exercises 19–24 represents the volume of a solid of revolution obtained by rotating a region around either the x- or y-axis. Find this region.

$π \int_{1}^{3} (x^{2} - 2 x + 1) dx$

Prove Theorem 6.22 by solving the initial-value problem $\frac{d T}{d t} = k (A - T)$ with T(0) = T0, where k and A are constants.

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The mass of the solid of revolution obtained by rotating the graph of y = 4.5 − 0.5x2 on [0, 3] around the y-axis and whose density at height y is ρ( y) = 1.3 − 0.233y ounces per cubic inch.

Short Answer

Step by step solution

Step1. Given Information

Step2. Calculate

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Most popular questions from this chapter

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The mass of the solid of revolution obtained by rotating the graph of y = 4.5 − 0.5 $x^{2}$ on [0, 3] around the y-axis and whose density at height y is ρ( y) = 1.3 − 0.233y ounces per cubic inch.