Chapter 6: Applications of Double Integration
Section 6.4: Average Value
Example 6.4.2
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.
Solution
Mathematical Solution
The average value of F=2 x2+3 y2 over the region shown in Figure 6.1.2(a) is
∫−3/21∫y23−y/22x2+3y2ⅆxⅆy∫−3/21∫y23−y/21ⅆxⅆy = 16875/1792125/48 = 405112
The numerator is the volume computed in Example 6.2.2, while the denominator is the area computed in Example 6.1.2.
Maple Solution - Interactive
A solution from first principles entails simply formulating and evaluating the integrals for volume and area as found in Example 6.2.2 and Example 6.1.2, respectively.
Solution from first principles
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫−3/21∫y23−y/22 x2+3 y2 ⅆx ⅆy∫−3/21∫y23−y/21 ⅆx ⅆy = 405112
Maple Solution - Coded
Use the FunctionAverage command in the Student MultivariateCalculus package
Student:-MultivariateCalculus:-FunctionAverage2 x2+3 y2,x=y2..3−y/2,y=−3/2..1,output=integral
∫−321∫y232−12y2x2+3y2ⅆxⅆy∫−321∫y232−12y1ⅆxⅆy
Student:-MultivariateCalculus:-FunctionAverage2 x2+3 y2,x=y2..3−y/2,y=−3/2..1
405112
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