Area, Moments, and Centers of Mass

University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
8
Area, Moments, and Centers of Mass
In this section, we show how to use double integrals to calculate the areas of bounded regions in the plane and to find the average value of a function of two variables. Then we study the physical problem of finding the center of mass of a thin plate covering a region in the plane.
Areas of Bounded Regions in the Plane
The area of a closed, bounded plane region R is
Moments and Centers of Mass for Thin Flat Plates
1- Mass and first moment formulas for thin plates covering a region R in the xy-plane
2- Second moment formulas for thin plates in the xy-plane
Moments of inertia (second moments):
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
9
EXAMPLE 5: A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate’s density at the point (x, y) is ?(x, y) = 6x + 6y + 6 . Find the plate’s mass, first moments, center of mass, moment of inertia, and radii of gyration about the coordinate axes.
Solution: We sketch the plate and put in enough detail to determine the limits of integration for the integrals we have to evaluate .
The plate’s mass is
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
10
moment of inertia about the x-axis is
Notice that we integrate y2 times density in calculating Ix and x2 times density to find Iy Since we know Ix and Iy we do not need to evaluate an integral to find Io ; we can use the equation I0 = Ix + Iy instead:
University of Babylon Lecture: Roaya Mahmood Jaleel
faculty of Engineering Subject: Mathematics
Department of Chemical Engineering Stage : 2nd stage
11
Centroids of Geometric Figures
EXAMPLE 6: Find the centroid of the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x2
Solution We sketch the region and include enough detail to determine the limits of Integration We then set ? equal to 1 .