rogers (grr459) – HW13 – rusin – (55220)
1
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±riday night (12/7) and then you’re done!
001
10.0 points
Evaluate the double integral
I
=
i i
A
4
dxdy
with
A
=
b
(
x, y
) : 2
≤
x
≤
8
,
5
≤
y
≤
7
B
by frst identiFying it as the volume oF a solid.
1.
I
= 52
2.
I
= 54
3.
I
= 46
4.
I
= 48
correct
5.
I
= 50
Explanation:
The value oF
I
is the volume oF the solid
below the graph oF
z
=
f
(
x, y
) = 4 and above
the region
A
=
b
(
x, y
) : 2
≤
x
≤
8
,
5
≤
y
≤
7
B
.
Since
A
is a rectangle, this solid is a box with
base
A
and height 4. Its volume, thereFore, is
given by
length
×
width
×
height
= (8

2)
×
(7

5)
×
4
.
Consequently,
I
= 48
.
keywords: volume, double integral, rectangu
lar region, rectangular solid
002
10.0 points
The graph oF the Function
z
=
f
(
x, y
) = 6

x
is the plane shown in
z
6
x
y
Determine the value oF the double integral
I
=
i i
A
f
(
x, y
)
dxdy
over the region
A
=
b
(
x, y
) : 0
≤
x
≤
6
,
0
≤
y
≤
4
B
in the
xy
plane by frst identiFying it as the
volume oF a solid below the graph oF
f
.
1.
I
= 73 cu. units
2.
I
= 72 cu. units
correct
3.
I
= 70 cu. units
4.
I
= 71 cu. units
5.
I
= 69 cu. units
Explanation:
The double integral
I
=
i i
A
f
(
x, y
)
dxdy
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2
is the volume of the solid below the graph of
f
having the rectangle
A
=
b
(
x, y
) : 0
≤
x
≤
6
,
0
≤
y
≤
4
B
for its base. Thus the solid is the wedge
z
6
6
x
y
(6
,
4)
and so its volume is the area of triangular
face multiplied by the thickness of the wedge.
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