Chapter 6: Applications of Double Integration
Section 6.4: Average Value
The average value of fx on the interval x∈a,b is given by 1b−a∫abfx dx.
Similarly, the average value of fx,y on a region R of area AR is given by
1AR∫∫Rfx,y dA = ∫∫Rfx,y dA∫∫R1 dA
In polar coordinates, the average value of Fr,θ on a region R of area AR is given by
1AR∫∫RFr,θ dA = ∫∫RFr,θ dA∫∫R1 dA
where now, dA includes ∂x,y∂r,θ=r, the absolute value of the Jacobian of the transformation from polar to Cartesian coordinates.
Figure 6.4.1 shows the average value of fx,y=x2+y2 on R, the first-quadrant portion of the unit circle.
The average value is given by
∫01∫01−x2x2+y2 dy dxπ/4 = π/8π/4=12
However, there is as much volume above the plane z=1/2 as there is below.
use plots in
Figure 6.4.1 Average value of x2+y2
The volume below z=1/2 is given by ∫01/21/4∫01/2−x2x2+y2 dy dx = π/16, exactly half the total volume. Hence, care must be taken in interpreting the meaning of the average value of a function of several variables.
Maple Tools for Calculating Average Value
The Student MultivariateCalculus package contains the FunctionAverage command that will construct and evaluate the average value of a function of two variables in either Cartesian or polar coordinates.
In addition, this command for functions of two variables has been implemented in two task templates, one for Cartesian coordinates and one for polar coordinates. In the Cartesian case, the task template iterates in the order dy dx; and in the polar case, in the order dr dθ.
Of course, there is always the option of implementing the relevant integrals from first principles using either the Int and int commands at top level, or the double-integral templates in the Calculus palette.
Find the average value of F=x y over R, the finite region bounded by the graph of y=x 1−x and the x-axis. See Example 6.2.1 and Example 6.1.1.
Find the average value of F=2 x2+3 y2 over R, the finite region bounded by the graphs of x=y2 and y=3−2 x. See Example 6.2.2 and Example 6.1.2.
Find the average value of F=2 x+3 y+1 over R, the region bounded by the graphs of fx=sinx and gx=sin2 x on 0≤x≤π. See Example 6.2.3 and Example 6.1.3.
Find the average value of F=3 x2+2 y2+1 over R, the region bounded by the graphs of fx=arctanx+1−1/2 and gx=sinx on the interval x∈0,π/2.
See Example 6.2.4 and Example 6.1.4.
Find the average value of F=2−y−1/22 over R, the region bounded by the graphs of 1, cosx, and y=x on 0≤x≤1. See Example 6.2.7 and Example 6.1.7.
Find the average value of Fr,θ=1+2 r cosθ+3 r sinθ over R, the interior of the loop of the 4-leaf rose r=cos2 θ that straddles the positive x-axis. See Example 5.7.1.
Find the average value of Fr,θ=4−3 r cosθ+5 r sinθ over R, the region that is inside the cardioid r=1+ cosθ but outside the circle r=1. See Example 5.7.2.
Find the average value of Fr,θ=r2 over R, the region that is inside the circle r=3 sinθ but outside the cardioid r=1+sinθ. See Example 5.7.3.
Find the average value of Fr,θ=r22 cos2θ+3 sin2θ over R, the region that is inside the circle r=3 cosθ but outside the cardioid r=1+cosθ. See Example 5.7.4.
Find the average value of Fr,θ=r+cosθ over R, the region that is inside the circle r=3 cosθ but outside the limaçon r=2−cosθ. See Example 5.7.5.
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