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Chapter 13: Q 46. (page 1039)

$C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$
If the density at each point in $C$ is proportional to the point’s distance from the $y$ -axis, find the mass of $C$ .

Short Answer

Expert verified

The mass of region is $m = \frac{4}{3} k$

Step by step solution

01

Given Information

We are given that $C = \{(x, y) ∣ x^{2} + y^{2} \leq 1\}$

02

Calculating Mass of Region

Mass of region is given by $m = \iint_{Ω} ρ (x, y) d A$

$ρ (x, y) = k x$

$m = \int_{- 1}^{1} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} k x d y d x$

According to figure

$m = 4 \times mass of a quadrant$

$m = 4 \int_{0}^{1} \int_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

$m = 4 \int_{0}^{1} \int_{0}^{\sqrt{1 - x^{2}}} k x d y d x$

Calculating inner integral first

$m = 4 \int_{01}^{1} k x [y]_{0}^{\sqrt{1 - x^{2}}} d x$

$m = 4 k \int_{0}^{1} x \sqrt{1 - x^{2}} d x$

Put $1 - x^{2} = t^{2}$

$\Rightarrow - 2 x d x = 2 t d t$

$m = 4 k \int_{0}^{1} t^{2} d t = 4 k {[\frac{t^{3}}{3}]}_{0}^{1} = 4 k [\frac{1}{3}]$

Mass of region is $m = \frac{4}{3} k$

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Most popular questions from this chapter

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$\int \int_{R} x \sin x \cos y d A, w h e r e R = {(x, y) | - 3 \leq x \leq 2 a n d - 2 \leq y \leq 2}$

Let $Ω$ be a lamina in the xy-plane. Suppose $Ω$ is composed of n non-overlapping laminæ role="math" localid="1650321722341" $Ω_{1}, Ω_{2}, \dots, Ω_{n} .$ Show that if the masses of these laminæ are $m_{1}, m_{2}, \dots, m_{n}$ and the centers of masses are $({\bar{x}}_{1}, {\bar{y}}_{1}), ({\bar{x}}_{2}, {\bar{y}}_{2}), \dots, ({\bar{x}}_{n}, {\bar{y}}_{n}),$ then the center of mass of $Ω$ is $(\bar{x}, \bar{y}),$ where

$\bar{x} = \frac{\sum_{k = 1}^{n} m_{k} {\bar{x}}_{k}}{\sum_{k = 1}^{n} m_{k}} and \bar{y} = \frac{\sum_{k = 1}^{n} m_{k} {\bar{y}}_{k}}{\sum_{k = 1}^{n} m_{k}} .$

In Exercises 61–64, let R be the rectangular solid defined by

$R = \{(x, y, z) | 0 \leq a_{1} \leq x \leq a_{2}, 0 \leq b_{1} \leq y \leq b_{2}, 0 \leq c_{2} \leq z \leq c_{2}\} .$

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2}) .$

(b) Verify that $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2})$ is the center of mass by using the appropriate integral expressions.

Evaluate the iterated integral :

$\int_{0}^{1} \int_{2}^{4} \int_{- 1}^{5} (x + y z^{2}) d x d y d z$

In Exercises 57–60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.

Assume that the density of R is uniform throughout, and find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

See all solutions

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