Problem 1

Show[
Plot[(Sqrt[4-(x^2)]),
{x,0,2},
PlotRange->{0,2},
AspectRatio->1,
ImageSize->400
],
DensityPlot[
x+y,
{x,0,2},{y,0,(Sqrt[4-(x^2)])},
ColorFunction->"Rainbow"
]
]

(∫_0^2)(∫_0^(Sqrt[4-(x^2)]))(x+y)\[DifferentialD]y\[DifferentialD]x

(∫_0^(π/2))(∫_0^2)(s Cos[ϕ]+s Sin[ϕ])s\[DifferentialD]s\[DifferentialD]ϕ

Problem 2

Show[
Plot[16-(x^2),
{x,-4.5,4.5},
PlotRange->{0,17},
AspectRatio->1,
ImageSize->400
],
DensityPlot[
y,
{x,-4,4},{y,0,16-(x^2)},
ColorFunction->"Rainbow"
]
]

(∫_(-4)^4)(∫_0^(16-(x^2)))k y\[DifferentialD]y\[DifferentialD]x

Problem 3

Show[
Plot[
6-x,
{x,-1,7},
PlotRange->{-1,7},
AspectRatio->1,
ImageSize->400
],
DensityPlot[
(x^2)+(y^2),
{x,0,6},{y,0,6-x},
ColorFunction->"Rainbow"
]
]

(∫_0^6)(∫_0^(6-x))k((x^2)+(y^2))\[DifferentialD]y\[DifferentialD]x

Problem 4

Plot3D[
{2x+3y+6,2x+7y+8},
{x,0,1},{y,0,1},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
AxesOrigin->{0,0,0},
ImageSize->600,
AxesStyle->Black,
Mesh->None,
PlotRange->{0,20}
]

(∫_0^1)(∫_0^1)(∫_(2x+3y+6)^(2x+7y+8))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

Problem 5

(∫_0^1)(∫_0^1)(∫_(2x+3y+6)^(2x+7y+8))k y\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

Problem 6

Plot3D[
{1-x-y,0},
{x,0,1},{y,0,1-x},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
AxesOrigin->{0,0,0},
ImageSize->600,
AxesStyle->Black,
Mesh->None,
PlotRange->{{-.2,1.2},{-.2,1.2},{0,1.2}}
]

V=(∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

{(((∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))x\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x)/V),(((∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))y\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x)/V),(((∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))z\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x)/V)}

{M=(∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))z\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x,(((∫_0^1)(∫_0^(1-x))(∫_0^(1-x-y))(z^2)\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x)/M)}

Problem 7

Show[
Plot[
(Sqrt[x]),
{x,0,2.2},
PlotRange->{0,2},
AspectRatio->Automatic,
ImageSize->400
],
RegionPlot[
y<(Sqrt[x]),
{x,0,2},{y,0,(Sqrt[2])},
PlotStyle->LightOrange
]
]

(∫_0^2)(∫_0^(Sqrt[x]))\[DifferentialD]y\[DifferentialD]x

Problem 8

Plot3D[
{(Sqrt[x-(y^2)]),-(Sqrt[x-(y^2)])},
{x,0,2},{y,-2,2},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
AxesOrigin->{0,0,0},
ImageSize->600,
AxesStyle->Black,
Mesh->None,
BoundaryStyle->None,
PlotRange->{-2,2}
]

(∫_0^2)(∫_(-(Sqrt[x]))^(Sqrt[x]))(∫_(-(Sqrt[x-(y^2)]))^(Sqrt[x-(y^2)]))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

4(∫_0^2)(∫_0^(Sqrt[x]))(∫_0^(Sqrt[x-(y^2)]))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

Problem 9

Plot3D[
{(Sqrt[4-(y^2)]),0},
{x,0,2},{y,x,2},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
AxesOrigin->{0,0,0},
ImageSize->600,
AxesStyle->Black,
Mesh->None,
PlotRange->{0,2}
]

(∫_0^2)(∫_0^y)(2/(Sqrt[4-(y^2)]))\[DifferentialD]x\[DifferentialD]y

(∫_0^2)(∫_x^2)(2/(Sqrt[4-(y^2)]))\[DifferentialD]y\[DifferentialD]x

Problem 10

Plot3D[
{1+(x^2)+(y^2),0},
{x,-1,1},{y,-(Sqrt[1-(x^2)]),(Sqrt[1-(x^2)])},
ColorFunction->"Rainbow",
SphericalRegion->True,
AspectRatio->1,
AxesOrigin->{0,0,0},
ImageSize->600,
AxesStyle->Black,
Mesh->None,
PlotRange->{0,2.5}
]

(∫_(-1)^1)(∫_(-(Sqrt[1-(x^2)]))^(Sqrt[1-(x^2)]))(∫_0^(1+(x^2)+(y^2)))\[DifferentialD]z\[DifferentialD]y\[DifferentialD]x

(∫_0^(2π))(∫_0^1)(∫_0^(1+(r^2)))r\[DifferentialD]z\[DifferentialD]r\[DifferentialD]θ