Chapter 13: Q 35. (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 square of the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .

Short Answer

Expert verified

The mass of lamina is $m = \frac{17}{6} k$ .

Step by step solution

Given Information

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

Calculating Mass of Lamina

Formula is $m = \iint_{Ω} ρ (x, y) d A$

Also density is $ρ (x, y) = k x^{2}$

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

Solving inner integral first

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

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

$m = 2 k \int_{1}^{2} (x^{3} - x^{2}) d x$

$m = 2 k {[\frac{x^{4}}{4} - \frac{x^{3}}{3}]}_{1}^{2}$

Putting limits

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

$m = 2 k [\frac{17}{12}]$

Hence, $m = \frac{17}{6} k$

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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 square of the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .

Short Answer

Step by step solution

Given Information

Calculating Mass of Lamina

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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 square of the point’s distance from the y-axis, find the mass of T2.

Short Answer

Step by step solution

Given Information

Calculating Mass of Lamina

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Geometry

Mechanics Maths

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 square of the point’s distance from the $y$ -axis, find the mass of $T_{2}$ .