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Chapter 13: Q 53. (page 1039)

Let $S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$
If the density at each point in S is proportional to the point鈥檚 distance from the y-axis, find the mass of S.

Short Answer

Expert verified

Answer is $\frac{2}{3} k, k$ is the constant of proportionality.

Step by step solution

01

Step 1. Given information

$S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$

02

Step 2. Explanation

$S = \{(x, y) | x^{2} + y^{2} 鈮� 1, x 鈮� 0\}$

Mass $= {鈭�}_{0}^{1} {鈭�}_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x$

$= 2 {鈭�}_{0}^{1} {鈭�}_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

Take $x = r \cos 胃, y = r \sin 胃$

role="math" localid="1653727977284" $\begin{array}{rcl} Mass & = & 2 {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} k r \cos 胃 r d r d 胃 \\ = & 2 k {鈭�}_{0}^{\frac{蟺}{2}} {鈭�}_{0}^{1} r^{2} \cos 胃 d r d 胃 \\ = & 2 k {鈭�}_{0}^{\frac{蟺}{2}} {[\frac{r^{3}}{3}]}_{0}^{1} \cos 胃 d 胃 \\ = & \frac{2}{3} k {[\sin 胃]}_{0}^{\frac{蟺}{2}} \\ = & \frac{2}{3} k [\sin (\frac{蟺}{2}) - \sin 0] \\ = & \frac{2}{3} k \end{array}$

Mass is equal to $\frac{2}{3} k, k$ is the constant of proportionality.

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

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