Chapter 13: Q 32. (page 1039)

Let $T_{2}$ be triangular region with vertices $(1, 0), (2, 1), and (2, - 1)$
If the density at each point in $T_{2}$ is proportional to the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .

Short Answer

Expert verified

The mass is $m = \frac{5}{2} k$

Step by step solution

Given Information

The vertices of triangular region is $(1, 0), (2, 1), and (2, - 1)$

Calculation of Mass

Mass of the lamina is calculated by

$m = \iint_{Ω} ρ (x, y) d A$

$ρ (x, y)$ is uniform density and is proportional to point's distance from $y$ axis.

$ρ (x, y) = k x$

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

Solving inner integral first

$m = \int_{1}^{2} k x [y]_{- x + 1}^{x - 1} d x$

$m = \int_{1}^{2} k x [x - 1 - (- x + 1)] d x$

$m = 2 k \int_{1}^{2} (x^{2} - x) d x$

$m = 2 k {[\frac{x^{3}}{3} - \frac{x^{2}}{2}]}_{1}^{2}$

$m = 2 k [(\frac{2^{3}}{3} - \frac{2^{2}}{2}) - (\frac{1}{3} - \frac{1}{2})]$

$m = 2 k [(\frac{8}{3} - \frac{4}{2}) - (\frac{1}{3} - \frac{1}{2})]$

$m = 2 k [\frac{5}{6}]$

Hence, $m = \frac{5}{3} k$

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Let $T_{2}$ be triangular region with vertices $(1, 0), (2, 1), and (2, - 1)$
If the density at each point in $T_{2}$ is proportional to the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .

Short Answer

Step by step solution

Given Information

Calculation of Mass

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Let T2be triangular region with vertices (1,0),(2,1), and(2,-1)If the density at each point in T2is proportional to the point’s distance from the y-axis, find the mass of T2.

Short Answer

Step by step solution

Given Information

Calculation of Mass

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Statistics

Discrete Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Let $T_{2}$ be triangular region with vertices $(1, 0), (2, 1), and (2, - 1)$
If the density at each point in $T_{2}$ is proportional to the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .