Problem

Find the coordinates of the centroids of the give…

Problem 31 Easy Difficulty

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. Find the location of the centroid of a hemisphere of radius $a$. Using this result, locate the centroid of the northern hemisphere of Earth for which the radius is $6370 \mathrm{km}$.

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Calculus 2 / BC

Basic Technical Mathematics with Calculus

Chapter 26

Applications of Integration

Section 4

Centroids

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Video Transcript

were given a region and were asked to find the Centro it of this region. The region is the quarter of the unit circle lying in the first quadrant. So first of all, thinking about symmetry here, notice that Oh, this region is symmetric about line lie equals X so follows that centrally lives on the line Y equals X in the first quadrant by the symmetry principles. This helps us simplify things so that we really only have to calculate one of the coordinates for the central. OId therefore really need to calculate one of the moments of this region. Now, before we do this, let's write an equation for a region. So the unit circle has equation X squared plus y squared equals one now because we're in the First Squadron have that X and Y are both going to be greater than equal to zero, he says. Solving for why, in terms of X, we get that. Why is going to be equal to the positive square root of one minus X squared? Excuse me. Now, using this, let's find the Y moment using equation one. So the wind moment is going to be density for a Central is one times the integral. And here because we're in the first quadrant, we're only going from X equals zero two X equals one of x times the function times one minus x squared d x And this is in this world that's easy to evaluate using substitution. So you would set, for example, of you to be one minus x squared. Then you'd have it do you would be negative two x dx replacing the X outside the X DX outside in terms of you and changing the parameters, we see that the little meat will be equal to one third after calculating. So we have the moment now the total mass of this region, all this is going to be the density of the region. Just one times the area. Yeah, So the area of the region is 1/4 of the area of the unit circles. This is going to be 1/4 times pie or pie A or four, and therefore it follows That s n Troy Eid is going to have coordinates. Am why over m followed by an X over m. So we were calculated. The N Y was one third and m this pile or four. This is going to be 12 of the pie. And moreover, by the cemetery principle, we know that I'm sorry. This is a mistake. Should be 4/3 pots by the symmetry principle. You know, the other coordinate is the same. So we have for over three pie. So our central it has coordinates four or three pie for over three pie.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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