A4-format til udskrift. - Aarhus Universitet

98 II. INTEGRATION
3.29. Kile ☞ [S] 12.3 Double integrals over general regions
Kile - figur
z
x
2
−2
D = {(x,y)| − 2 ≤ x ≤ 2,0 ≤ y ≤ 4 − x 2 }
E = {(x,y,z)|(x,y) ∈ D,0 ≤ z ≤ 1
2 y}
3.30. Volumen af kile ☞ [S] 12.3 Double integrals over general regions
Kile - fortsat
D = {(x,y)| − 2 ≤ x ≤ 2,0 ≤ y ≤ 4 − x 2 }
er en Type I mængde.
Volumenet af kilen er
V =
D
2
1
y dA =
2 −2
√ 4−x2 1
y dy dx
0 2
3.31. Type I ☞ [S] 12.3 Double integrals over general regions
Kile - fortsat
2
1
V = ydA =
D 2 −2
√ 4−x2 1
ydy dx
0 2
2
1
=
4 y2
√ y= 4−x2 dx
−2
2
y=0
1
=
−2 4 (4 − x2 )dx
= x − 1
12 x3
2 = 8
3
−2
3.32. Regneregler
Regneregler for dobbeltintegral
☞ [S] 12.3 Double integrals over general regions
6
(f(x,y) + g(x,y))dA =
D
f(x,y)dA + g(x,y)dA
7
D
D
cf(x,y)dA = c
D
2
D
y
f(x,y)dA