Chapter 13: Q.24 (page 991)

In Exercises $24 - 30$ , let role="math" localid="1650649060037" $T$ be the triangular region with vertices $(0, 0), (1, 1)$ ,and $(1, - 1)$ .
Find the centroid of role="math" localid="1650649066645" $T$ .

Short Answer

Expert verified

Thus, the centroid of the triangular region is $\bar{x} = \frac{3}{4}, \bar{y} = 0$

Step by step solution

Given information

Centroid of a plane figure is defined as the point of intersection of medians.

Calculation

For example, take a triangle with vertices $(0, 0), (1, 1),$ and $(1, - 1)$ .

Use the formula for centroid

$\bar{x} = \frac{\iint_{Ω} x ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A} and \bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A}$

ρ (x, y)

is the uniform density of the lamina. If

ρ (x, y)

is proportional to the point's distance from the y - axis.

$ρ (x, y) = k x$

$\bar{x} = \frac{\int_{0}^{1} \int_{- x}^{x} x k x d y d x}{\int_{0}^{1} \int_{- x}^{x} k x d y d x}$

localid="1650649551169" $\bar{x} = \frac{\int_{0}^{1} \int_{- x}^{x} k x^{2} d y d x}{\int_{0}^{1} \int_{- x}^{x} k x d y d x}$

$\bar{x} = \frac{\int_{0}^{1} k x^{2} [y]_{- x}^{x} d x}{\int_{0}^{1} k x [y]_{- x}^{x} d x} \bar{x} = \frac{\int_{0}^{1} k x^{2} [2 x] d x}{\int_{0}^{1} k x [2 x] d x} \bar{x} = \frac{\int_{0}^{1} k x^{3} d x}{\int_{0}^{1} k x^{2} d x} \bar{x} = \frac{k {[\frac{x^{4}}{4}]}_{0}^{1}}{k {[\frac{x^{3}}{3}]}_{0}^{1}} \bar{x} = \frac{3}{4}$

Now

$\bar{y} = \frac{\iint_{0} y ρ (x, y) d A}{\iint_{0} ρ (x, y) d A} \bar{y} = \frac{\int_{0}^{1} \int_{- x}^{x} y k x d y d x}{\int_{0}^{1} \int_{- x}^{x} k x d y d x} \bar{y} = \frac{\int_{0}^{1} k x {[\frac{y^{2}}{2}]}_{- x}^{x}}{\int_{0}^{1} k x [y]_{- x}^{x} d x} \bar{y} = \frac{\int_{0}^{1} k x [0] d x}{\int_{0}^{1} 2 k x^{2}} \bar{y} = \frac{\int_{0}^{1} k x [0] d x}{k {[\frac{2}{3} x^{3}]}_{0}^{1}} \bar{y} = \frac{\frac{2}{3} k}{\int_{0}} \bar{y} = 0$

Thus, the centroid of the triangular region is $\bar{x} = \frac{3}{4}, \bar{y} = 0$

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In Exercises $24 - 30$ , let role="math" localid="1650649060037" $T$ be the triangular region with vertices $(0, 0), (1, 1)$ ,and $(1, - 1)$ .
Find the centroid of role="math" localid="1650649066645" $T$ .

Short Answer

Step by step solution

Given information

Calculation

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In Exercises 24-30, let role="math" localid="1650649060037" Tbe the triangular region with vertices (0,0),(1,1),and(1,-1).Find the centroid of role="math" localid="1650649066645" T.

Short Answer

Step by step solution

Given information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Pure Maths

Geometry

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises $24 - 30$ , let role="math" localid="1650649060037" $T$ be the triangular region with vertices $(0, 0), (1, 1)$ ,and $(1, - 1)$ .
Find the centroid of role="math" localid="1650649066645" $T$ .