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Q11E

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Found in: Page 740

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# Determine the solid described by the given inequalities.

The outline sketch of the given region is given below in the Figure:

See the step by step solution

## Step 1: Given data

The given inequalities are $$2 \le \rho \le 4,0 \le \phi \le \frac{\pi }{3},0 \le \theta \le \pi$$.

## Step 2: Concept of graph

Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them.

## Step 3: Simplify the expression

Convert the equation $$\rho = 2$$ to the following equation as shown below.

$$\begin{array}{c}\rho = 2\\\sqrt {{x^2} + {y^2} + {z^2}} = 2\\{x^2} + {y^2} + {z^2} = 4\end{array}$$

Also convert the equation $$\rho = 4$$ to the following equation as shown below.

$$\begin{array}{c}\rho = 4\\\sqrt {{x^2} + {y^2} + {z^2}} = 4\\{x^2} + {y^2} + {z^2} = 16\end{array}$$

From the equations above, it is identified that the required region lies between two spheres of radii 1 and 2 and both centered at origin. Since $$\theta$$ varies from 0 to $$\pi$$ and since $$\phi$$ varies from 0 to $$\frac{\pi }{3}$$, only the region present right to the $$xz$$-plane and the region present above the curvature $$\phi = \frac{\pi }{3}$$ is the required region.

Thus, the outline sketch of the given region is given below in the Figure: