Wydział: WILiŚ, Budownictwo i Transport, sem.2
dr Jolanta Dymkowska
Całka podwójna - ciąg dalszy
Zad.1 Wprowadzając współrzędne biegunowe obliczyć:

1.1 (x2 + y2) dxdy D : x2 + y2 4
D
Å„Å‚
òÅ‚

x2 + y2 25
1.2 x2 + y2 - 9 dxdy D :
ół
D
x2 + y2 9
Å„Å‚
òÅ‚
2 x2 + y2 2x
y
1.3 dxdy D :
x
ół
D
x2 + y2 1

1.4 (x + y) dxdy D : x2 + y2 x + y
D
Å„Å‚
òÅ‚
x2 + y2 25
1.5 ln(1 + x2 + y2) dxdy D :
ół
D
x 0
Å„Å‚
òÅ‚

x2 + y2 1
1.6 x2 x2 + y2 dxdy D :
ół
D
y x
Å„Å‚
òÅ‚

x2 + y2 2x
1.7 4 - x2 - y2 dxdy D :
ół
D
y 0
Å„Å‚
ôÅ‚
ôÅ‚
x2 + y2 1
ôÅ‚
òÅ‚

1
"
1.8 dxdy D :
x2 + y2 4
x2+y2
ôÅ‚
D ôÅ‚
ôÅ‚
ół
y x
Zad.2 Za pomocą całki podwójnej obliczyć pole obszaru ograniczonego liniami:
2.1 y2 = 4 + x, x + 3y = 0
2.2 y = ln x, y = x - 1, y = -1
2.3 x = 4 - y2, x = y - 2
Ä„
2.4 y = arctg x, x = 0, x = 1, y =
2
2.5 y = sin x, y = - sin x, x " [0, Ä„]
"
2.6 x2 + y2 = 2y, x2 + y2 = 4y, y = x, y = 3 x
2.7 (x2 + y2)2 = 2(x2 - y2), x2 + y2 = 2x
2.8 (x2 + y2)2 = x2
Zad.3 Za pomocą całki podwójnej obliczyć objętość bryły ograniczonej powierzchniami:
3.1 y = x2, y = 1, z = 0, z = 2y
3.2 x = 0, y = 0, x + y = 1, z = 0, z = 2xy
3.3 y2 + x = 0, z2 + x = 0, x + 4 = 0
3.4 x = 0 y = 1, 2x + y = 5, z = 0, z = xy
"
3.5 x = 1, x = 4, y = 0, y = x, z = 0, z = xy
"
3.6 y = x2, y = 2x2, y = 1, z = 0, z = 4 x
3.7 y2 - x + 1 = 0, x - y = 3, z = 0, z = 2x + y
3.8 x = y2, y = x - 2, z = x + y + 2, z = 0
3.9 x = 1, x = e, y = 1 - ln x, y = 2 + 2 ln x, z = 0, z = x + 2y
"
3.10 x = 1, y = 0, y = x, z = 0, z = arctg y
3.11 x = 0, y = 0, x + y = 4, z = 0, z = 1 + x2 + y2
3.12 y = x, y = x2, z = 0, z = x2 + y2
3.13 x = 0, y = 2x, y = 1, z = 0, z = -x2 - y2
3.14 z = 0, z = 3 - x, y2 = 3x
3.15 y = ex, y = e2x, y = e, z = x, z = 2x
3.16 x2 + y2 = 1, z = 0, z = 6x2 + 2y2
3.17 x2 + y2 = 4, z = 0, z = x + 2y + 8
3.18 x2 + y2 = 4, z = x, z = x + 1
3.19 x2 + y2 = z, z = 8 - x2 - y2
3.20 x2 + y2 = 1, z = x - 1, z = x2
3.21 x2 + y2 + z2 = 8, x2 + y2 = z2

3.22 z = 9 x2 + y2, z = 22 - x2 - y2

3.23 z = 36 - x2 - y2, 9z = x2 + y2
3.24 x2 + y2 = z, x2 + y2 = x, x2 + y2 = 2x, z = 0
3.25 x2 + y2 = 4, z = 0, 4z = 16 - x2 - y2
3.26 x2 + y2 = 2x, z2 = 4x
3.27 x2 + y2 = z, z = x + y
3.28 z = 2 - 10(x2 + y2), z = 2 + 20y
3.29 x2 + y2 = 4z2, x2 + y2 = 8z
3.30 x2 + y2 = -4y, z = 0, z = (y - x)2
1
3.31 x2 + y2 = 1, z = x + y, z = 0, (dla x, y 0)
9
3.32 25x2 + 9y2 = z, z = 4
1
3.33 x2 + y2 = 1, z = 2xy, z = 0, (dla x, y 0)
16
1 1
3.34 x2 + y2 = 1, z + x = 6, z = 0
9 4
1
1
16
3.35 x2 + y2 = 1, z = e-( x2+y2), z = 0
16