16 MULTIPLE INTEGRALS
16 MULTIPLE INTEGRALS
16 MULTIPLE INTEGRALS
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
CHAPTER <strong>16</strong><br />
<strong>MULTIPLE</strong> <strong>INTEGRALS</strong><br />
WORKSHOP/DISCUSSION<br />
• Define the centroid (x, y) of a plane region R as the center of gravity, obtained by using a density of 1 for<br />
the entire region. So x = 1 ∫∫<br />
xdxdyand y = 1 ∫∫<br />
ydxdy. Show the students how to find<br />
A (R)<br />
A (R)<br />
the centroid for two or three figures like the following:<br />
x=2-y@<br />
y=2-x@<br />
2/3<br />
_1/Ï3<br />
1/Ï3<br />
_1/3<br />
Show the students that if x = 0isanaxisofsymmetryforaregionR, thenx = 0, and more generally,<br />
that (x, y) is on the axis of symmetry. Point out that if there are two axes of symmetry, then the centroid<br />
(x, y) is at their intersection.<br />
• Consider the triangular region R shown below, and assume that the density of an object with shape R is<br />
proportional to the square of the distance to the origin. Set up and evaluate the mass integral for such an<br />
object, and then compute the center of mass (x, y).<br />
900
Change language