The average value or mean value of a continuous function f x , y R = a , b × c , d is defined as f ave = 1 A R ∬ 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 ≤ x ≤ 1 and 0 ≤ y ≤ 2.
The average value or mean value of a continuous function f x , y R = a , b × c , d is defined as f ave = 1 A R ∬ 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 ≤ x ≤ 1 and 0 ≤ y ≤ 2.
The average value or mean value of a continuous function
f
x
,
y
R
=
a
,
b
×
c
,
d
is defined as
f
ave
=
1
A
R
∬
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
≤
x
≤
1
and
0
≤
y
≤
2.
A contour map is shown for a function f on the square R = [0,4 x [0, 4]. Use midpoints
with 4 total rectangles to estimate the value of || f(x,y) dA.
R
y 4
4
10
0 0
10 20 30
2
10
20
30
4 x
2.
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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