Chapter 13: Q 28 (page 1039)

Let $T$ be triangular region with vertices $(0, 0), (1, 1), and (1, - 1)$
If the density at each point in $T$ is proportional to the point’s distance from the $x$ -axis, find the mass of $T$ .

Short Answer

Expert verified

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

Step by step solution

Given Information

We need to determine mass of triangular region.

Density at each point is proportional to point's distance from $x$ axis

$ρ (x, y) = k y$

Calculation of Mass

Mass is symmetric about $x$ axis.

Hence, total mass is twice of mass of upper half triangle.

$\Rightarrow m = 2 \int_{0}^{1} \int_{0}^{x} k y d y d x [m = \iint_{Ω} ρ (x, y) d A]$

Solving inner integral first

$m = 2 k \int_{0}^{1} {[\frac{y^{2}}{2}]}_{0}^{x} d x$

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

$m = k {[\frac{x^{3}}{3}]}_{0}^{1}$

Solving further

$m = \frac{k}{3}$

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Let $T$ be triangular region with vertices $(0, 0), (1, 1), and (1, - 1)$
If the density at each point in $T$ is proportional to the point’s distance from the $x$ -axis, find the mass of $T$ .

Short Answer

Step by step solution

Given Information

Calculation of Mass

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Let Tbe triangular region with vertices (0,0),(1,1),and(1,-1)If the density at each point in Tis proportional to the point’s distance from the x-axis, find the mass of T.

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

Pure Maths

Decision Maths

Applied Mathematics

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let $T$ be triangular region with vertices $(0, 0), (1, 1), and (1, - 1)$
If the density at each point in $T$ is proportional to the point’s distance from the $x$ -axis, find the mass of $T$ .