3540 The average value or mean value of a continuous functionf(x, y) over a rectangle R

Chapter 14, Problem 36

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The average value or mean value of a continuous function $f(x, y)$ over a rectangle $R=[a, b] \times[c, d]$ is defined as

$f_{\mathrm{ave}}=\frac{1}{A(R)} \iint_{R} f(x, y) d A$

where $A(R)=(b-a)(d-c)$ is the area of the rectangle $R$ (compare to Definition 5.8.1). Use this definition in these exercises.

Find the average value of $f(x,y)=x^{2}+7y $ over the rectangle ${[0,3] \times[0,6]}$.

Equation Transcription:

$f(x,\ y)$

$R=[a,\ b]\ \times{}[c,\ d]$

${f}_{ave}=\frac{1}{A(R)}\int\limits_{\ }^{\ }\int\limits_{R}^{\ }f\ (x,\ y)\ d\ A$

$A(R)=(b-a)(d-c)$

$R$

$\ f(x,y)={x}^{2}+7y$

$[0,3]\times{}[0,6]$

Text Transcription:

f(x, y)

R=[a, b]x [c, d]

f_ave=1/A(R) integral integral_R f (x, y) d A

A(R)=(b-a)(d-c)

f(x,y)=x^2+7y

[0,3]x[0,6]

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