Chapter 6: Applications of Double Integration
Section 6.4: Average Value
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.
The average value of F=x y over the region shown in Figure 6.1.1(a) is
∫01∫0x⁢1−xx⁢yⅆyⅆx∫01∫0x⁢1−x1ⅆyⅆx = 1/1201/6 = 120
The numerator is the volume computed in Example 6.2.1, while the denominator is the area computed in Example 6.1.1.
Maple Solution - Interactive
The task template in Table 6.4.1(a) implements, for double integrals (in Cartesian coordinates) that can be iterated in the order dy dx, the FunctionAverage command in the Student MultivariateCalculus package.
Calculus - Multivariate≻Integration≻Average Value≻Cartesian 2-D
Average Value of a Function: fx,y
Inert integral: dy dx
Table 6.4.1(a) Task template implementing the FunctionAverage command
The task template in Table 6.4.1(a) displays the ratio of integrals needed in the computation of the average value. Hence, a solution from first principles entails simply formulating and evaluating these integrals.
Solution from first principles
Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫01∫0x 1−xx y ⅆy ⅆx∫01∫0x 1−x1 ⅆy ⅆx = 120
Maple Solution - Coded
Use the FunctionAverage command in the Student MultivariateCalculus package
Student:-MultivariateCalculus:-FunctionAveragex y,y=0..x 1−x,x=0..1,output=integral
Student:-MultivariateCalculus:-FunctionAveragex y,y=0..x 1−x,x=0..1
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