Chapter 6: Applications of Double Integration
Section 6.4: Average Value
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.
The average value of F=2−y−1/22 over the region shown in Figure 6.1.7(a) is
∫Q1∫arccosyy2⁢x2+3⁢y2ⅆxⅆy∫Q1∫arccosyy1 ⅆxⅆy = 0.18124033810.09951138789 = 1.821302486
where Q≐0.74 is the solution of the equation x=cosx. (Recall that point P in Figure 6.1.7(a) has coordinates Q,Q because it is on the line y=x.) The numerator is the volume computed in Example 6.2.7, while the denominator is the area computed in Example 6.1.7.
Maple Solution - Interactive
A solution from first principles entails simply formulating and evaluating the integrals for volume and area as found in Example 6.2.7 and Example 6.1.7, respectively.
Context Panel: Assign Name
Solve cosx=x for Q
Write the equation cosx=x.
Context Panel: Solve≻Numerically Solve
Context Panel: Assign to a Name≻Q
cosx=x→solve0.7390851332→assign to a nameQ
Iterate in the order dx dy via the template in the Calculus palette
Iterated double-integral template
Context Panel: Evaluate and Display Inline
∫Q1∫arccosyyF ⅆx ⅆy∫Q1∫arccosyy1 ⅆx ⅆy = 1.821302486
Maple Solution - Coded
Solve the equation cosx=x for Q
Apply the fsolve command.
Obtain the average value of F
Use the FunctionAverage command in the Student MultivariateCalculus package
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