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

Chapter 14, Problem 39

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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.

Suppose that the temperature in degrees Celsius at a point \((x, y)\) on a flat metal plate is \(T(x, y)=10-8 x^{2}-2 y^{2}\), where \(x\) and \(y\) are in meters. Find the average temperature of the rectangular portion of the plate for which \(0 \leq x \leq 1\) and \(0 \leq y \leq 2\)

Equation Transcription:

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)

R

(x, y)

T(x, y)=10-8x^2-2y^2

x

y

0 leq x leq 1

0 leq  y leq  2