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Chapter 13: Q. 20 (page 1038)

Explain why the location of the centroid relates only to the geometry of the region and not its mass.

Short Answer

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

Step by step solution

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The location of the centroid relates only to the geometry of the region and not it's mass.

Calculation

For an example, take a triangle $A B C$ with vertices $(1, 1), (2, 0), a n d (2, 3)$ and join them. The point of intersection of medians $A D, B E$ and $C F$ is $G$

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Most popular questions from this chapter

Let $f (x, y, z)$ be a continuous function of three variables, let $Ω_{x y} = {(x, y) | a \leq x \leq b a n d h_{1} (x) \leq y \leq h_{2} (x)}$ be a set of points in the $x y$ -plane, and let $Ω = {(x, y, z) | (x, y) \in Ω_{x y} a n d g_{1} (x, y) \leq z \leq g_{2} (x, y)}$ be a set of points in 3-space. Find an iterated triple integral equal to the the triple integral $∭_{Ω} f (x, y, z) d V$ . How would your answer change if $Ω_{x y} = {(x, y) | a \leq y \leq b a n d h_{1} (y) \leq x \leq h_{2} (y)}$ ?

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$\int \int_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{π}{4} \leq x \leq \frac{π}{2} a n d \frac{π}{2} \leq y \leq π\}$

Evaluate each of the double integrals in Exercises $37 - 54$ as iterated integrals.

localid="1650380493598" $\int \int_{R} \sin (2 x + y) d A,$

wherelocalid="1650380496793" $R = \{(x, y) | 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq \frac{π}{2}\}$

Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

$∭_{Ω} z ρ (x, y, z) d V$

In Exercises 57–60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2}.

Assume that the density at each point in Ris proportional to the distance of the point from the xy-plane.

(a) Without using calculus, explain why the x- and y-coordinates of the center of mass are $\bar{x} = 2 and \bar{y} = \frac{3}{2},$ respectively.

(b) Use an appropriate integral expression to find the z-coordinate of the center of mass.

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Explain why the location of the centroid relates only to the geometry of the region and not its mass.

Short Answer

Step by step solution

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Calculation

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