A4-format til udskrift. - Aarhus Universitet

3. GENERELLE OMRÅDER 93
D = {(x,y)| − 1 ≤ x ≤ 1,2x 2 ≤ y ≤ 1 + x 2 }
3.12. Type I ☞ [S] 12.3 Double integrals over general regions
Eksempel 1 -
fortsat
1 2
1+x
(x + 2y)dA = (x + 2y)dy dx
D
=
=
=
−1 2x2 1 2
xy + y
−1
y=1+x 2
y=2x2 dx
1
(x(1 + x
−1
2 ) + (1 + x 2 ) 2 − x(2x 2 ) − (2x 2 ) 2 )dx
1
(−3x
−1
4 − x 3 + 2x 2 + x + 1)dx
1
= −3 x5 x4 x2
− + 2x3 + + x
5 4 3 2
= 32
15
3.13. Type II ☞ [S] 12.3 Double integrals over general regions
Figur
y
d
−1
x = h1(y) x = h2(y)
c
Område af Type II
D = {(x,y)|c ≤ y ≤ d,h1(y) ≤ x ≤ h2(y)}
3.14. Type II ☞ [S] 12.3 Double integrals over general regions
Type II integral
For f givet på
4 D = {(x,y)|c ≤ y ≤ d,h1(y) ≤ x ≤ h2(y)}
er integralet et itereret integral
5
D
d h2(y)
f(x,y)dA = f(x,y)dx dy
c
h1(y)
3.15. Type II ☞ [S] 12.3 Double integrals over general regions
Eksempel 2
x