Chapter 13: Q 38. (page 1039)

Let $R$ be rectangular region with vertices $(0, 0), (b, 0), (0, h), and (b, h)$
Find the centroid of $R$

Short Answer

Expert verified

The centroid is $\bar{x} = \frac{b}{2}, \bar{y} = \frac{h}{2}$

Step by step solution

Given Information

Vertices of rectangular region is $(0, 0), (b, 0), (0, h) and (b, h)$

Calculation of x-

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

Density is uniform, $ρ (x, y) = 1$

$\bar{x} = \frac{\int_{0}^{b} \int_{0}^{h} x d y d x}{\int_{0}^{b} \int_{0}^{h} d y d x}$

Solving inner integral first

$\bar{x} = \frac{\int_{0}^{b} x [y]_{0}^{h} d x}{\int_{0}^{b} [y]_{0}^{h} d x}$

$\bar{x} = \frac{\int_{0}^{b} x d x}{\int_{0}^{b} d x}$

Using limits $\bar{x} = \frac{\int_{0}^{b} x d x}{\int_{0}^{b} d x}$

Solving further

$\bar{x} = \frac{{[\frac{x^{2}}{2}]}_{0}^{b}}{[x]_{0}^{b}} = \frac{[\frac{b^{2}}{2}]}{[b]}$

$\bar{x} = \frac{b}{2}$

Calculation of y-

Similarly

$\bar{y} = \frac{\iint_{Ω} y ρ (x, y) d A}{\iint_{Ω} ρ (x, y) d A} = \frac{\int_{0}^{b} \int_{0}^{h} y d y d x}{\int_{0}^{b} \int_{0}^{h} d y d x} = \frac{\int_{0}^{b} {[\frac{y^{2}}{2}]}_{0}^{h} d x}{\int_{0}^{b} [y]_{0}^{h} d x} = \frac{\int_{0}^{b} [\frac{h^{2}}{2}] d x}{\int_{0}^{b} [h] d x}$

$\bar{y} = \frac{h \int_{0}^{b} x d x}{2 \int_{0}^{b} x d x} = \frac{h {[\frac{x^{2}}{2}]}_{0}^{b}}{2 {[\frac{x^{2}}{2}]}_{0}^{b}} = \frac{h [\frac{b^{2}}{2}]}{2 [\frac{b^{2}}{2}]}$

$\bar{y} = \frac{h}{2}$

Hence, centroid is $\bar{x} = \frac{b}{2}, \bar{y} = \frac{h}{2}$

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Let $R$ be rectangular region with vertices $(0, 0), (b, 0), (0, h), and (b, h)$
Find the centroid of $R$

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

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Let Rbe rectangular region with vertices (0,0),(b,0),(0,h),and(b,h)Find the centroid ofR

Short Answer

Step by step solution

Given Information

Calculation of x-

Calculation of y-

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Decision Maths

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Let $R$ be rectangular region with vertices $(0, 0), (b, 0), (0, h), and (b, h)$
Find the centroid of $R$