2. find the average value of the given function, f, over the given region, d. use your computations from the…

2. find the average value of the given function, f, over the given region, d. use your computations from the previous problem to your life easier! note that the formula gives you the “obvious” answer for (a), (b), and (e).\n(a) ( f(x,y)=1 ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(b) ( f(x,y)=3 ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(c) ( f(x,y)=2x ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(d) ( f(x,y)=x^{2}y^{3} ), ( d={(x,y):3leq xleq6,4leq yleq8} ).\n(e) ( f(x,y,z)=5 ),

2. find the average value of the given function, f, over the given region, d. use your computations from the previous problem to your life easier! note that the formula gives you the “obvious” answer for (a), (b), and (e).\n(a) ( f(x,y)=1 ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(b) ( f(x,y)=3 ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(c) ( f(x,y)=2x ), ( d={(x,y):0leq xleq3,0leq yleq2} ).\n(d) ( f(x,y)=x^{2}y^{3} ), ( d={(x,y):3leq xleq6,4leq yleq8} ).\n(e) ( f(x,y,z)=5 ),

Answer

Explanation:

Step1: Recall the formula for the average value of a function over a region

The formula for the average value of a function (f(x,y)) over a region (D) (where (D) is a rectangle ({(x,y):a\leq x\leq b,c\leq y\leq d})) is (\bar{f}=\frac{1}{(b - a)(d - c)}\iint_{D}f(x,y)dA)

For part (a), (a = 0,b = 3,c = 0,d = 2) and (f(x,y)=1)

Step2: Calculate the double - integral

(\iint_{D}1dA=\int_{0}^{3}\int_{0}^{2}1dydx)

First, integrate with respect to (y): (\int_{0}^{3}\left[y\right]{0}^{2}dx=\int{0}^{3}(2 - 0)dx)

Then integrate with respect to (x): (\int_{0}^{3}2dx=2\left[x\right]_{0}^{3}=2(3 - 0)=6)

Step3: Calculate the area of the region (D)

The area of the rectangle (D) is ((3 - 0)\times(2 - 0)=6)

Step4: Calculate the average value

Using the formula (\bar{f}=\frac{1}{(3 - 0)(2 - 0)}\iint_{D}1dA)

Substitute the values: (\bar{f}=\frac{6}{6}=1)

Answer:

The average value of (f(x,y) = 1) over the region (D={(x,y):0\leq x\leq3,0\leq y\leq2}) is (1)