Calculus II

Lesson 7a:  Centroids

First moments, centroids Papus' theorem.

       If we restrict the concept of center of gravity or center of mass to a closed plane curve we obtain the idea of  "centroid".  The centroid is that point on which a thin sheet matching the closed curve could be balanced. It is the "center of mass". Consider the portion of the parabola described by: , which lies above the x axis.  Since we are dealing with a plane surface the density is constant and need not concern us.  In addition, the centroid is a geometrically defined point  whose location is coordinate independent.

      Divide the area into strips of width   and measure the distance from the center of the strip,  , to the y axis.  If we approximate the area of each strip as , then the first moment   is defined as  , where n is the number of strips.

       In the limit as ->0, this becomes . The first moment about the y axis is therefore  Therefore, the x coordinate of the centroid is found to be .  To find the y coordinate of the centroid, observe that for an infinitesimally wide strip the centroid will lie one half the distance between the top and bottom of the strip.  Therefore,  

Ex. 1

   Locate the centroid of the plane area enclosed between the curve    , and between the y axis and the line x=3.

Warning, the name changecoords has been redefined

 The centroid lies on the line of symmetry of the surface area as could be anticipated.  Now we add a slight complication.

Ex. 2

We plot sin(x) and a circle centered at ( )  with radius 1 and locate the centroid of the resulting figure.

Warning, the name changecoords has been redefined

Now we will find the centroid of the area enclosed by the two plots.  First we note that the area we are interested in is described by   f(x)-g(x).  The center of each strip is    above the x axis.  The length of each strip is  f(x)-g(x) .

The formulas for the centroid therefore require modification to:

        =

First we find the limits of integration by finding the points of intersection of the two curves.

This certainly looks about right.

Pappus' Theorem

        Given a closed curve with area   A,  perimeter P and centroid  { }, and a line external to the closed curve whose distance from the centroid is d  , we rotate the plane curve around the line obtaining a solid of revolution. The volume of the solid is  , and the surface area is .

Ex. 3

Given the ellipse:  .  Find the surface area of the solid of revolution about the x axis. We rotate the ellipse around the x axis.  

Warning, the name changecoords has been redefined

plot3d

          Since, by symmetry, we know that the centroid of the ellipse is at (5,7) and the area of an ellipse is   (semi-major axis) x (semi-minor axis),  we immediately have, for the volume of the resulting solid;

Ex. 4

Given the parabolic arc ,  and the line y=10.  We wish to know the volume of the solid of revolution obtained by rotating this area around a line parallel to the x axis and 5 units from the lowest point on the arc.  For this problem we only need the y coordinate of the centroid.

        First we need to know the limits of integration.

Now compute the y coordinate of the centroid.

Now we need the distance of the centroid from the lowest point on the curve, which occurs at    .

The required volume is:

This is what this solid looks like.

Practice

1.  Find the centroid of each of the following figures.

       a.  The triangle formed by the x axis, the y axis and the line  

       b.  The area enclosed by the x axis, the y axis and the curve

       c.  The area enclose by the curves:  and

2.  An equilateral triangle, 2 units on each side, is rotated around a line parallel to,  and 2 units from,  one side.  Find the surface area and the volume of  the resulting solid.