Michael Corral: Vector Calculus

3.2 Double Integrals Over a General Region 107
Example 3.5. Find the volume V of the solid bounded by the three coordinate planes
and the plane 2x+y+4z=4.
4
y
(0,0,1)
z
2x+y+4z=4
y=−2x+4
x
(2,0,0)
0 (0,4,0)
y
0
R
2
x
(a)
(b)
Figure 3.2.4
Solution: ThesolidisshowninFigure3.2.4(a)withatypicalverticalslice. Thevolume
V is given by f(x,y)dA, where f(x,y)=z= 1 4
(4−2x−y) and the region R, shown in
R
Figure 3.2.4(b), is R={(x,y) : 0≤ x≤2, 0≤y≤−2x+4}. Using vertical slices in R gives
1
V=
4 (4−2x−y)dA
=
=
=
R
∫ 2
[∫ −2x+4
0
∫ 2
0
∫ 2
0
]
1
4 (4−2x−y)dy dx
0
(− 1 8 (4−2x−y)2∣ ∣y=−2x+4)
∣∣ dx
1
8 (4−2x)2 dx
y=0
=− 1 48 (4−2x)3∣ ∣2
∣∣ = 64
0 48 = 4 3
For a general region R, which may not be one of the types of regions we have considered
so far, the double integral R
f(x,y)dA is defined as follows. Assume that f(x,y)
is a nonnegative real-valued function and that R is a bounded region in 2 , so it can
be enclosed in some rectangle [a,b]×[c,d]. Then divide that rectangle into a grid of
subrectangles. Only consider the subrectangles that are enclosed completely within
the region R, as shown by the shaded subrectangles in Figure 3.2.5(a). In any such
subrectangle [x i ,x i+1 ]×[y j ,y j+1 ], pick a point (x i∗ ,y j∗ ). Then the volume under the surface
z= f(x,y) over that subrectangle is approximately f(x i∗ ,y j∗ )∆x i ∆y j , where∆x i = x i+1 − x i ,