
MTH243 (Calculus
for Functions of Several Variables)
MATLAB. Chapter 16:
Integrating Functions of Several Variables
Vladimir A. Dobrushkin,Lippitt
Hall 202C, 8745095,dobrush@uri.edu
In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI.
The Mathematica projects are created to help you learn new concepts. Mathematica is very useful in visualizing
graphs and surfaces in three dimensions. Matlab (commercial software) is also available at engineeering labs. Its
free version is called Octave. A student can also use free CASs: SymPy (based on Python), Maxima, or Sage.

 Chapter 12 Chapter 12: Functions of Several Variables
 Chapter 13 Chapter 13: Vectors
 Chapter 14 Chapter 14: Differentiating Functions
 Chapter 15 Chapter 15: Optimization
 Chapter 16 Chapter 16: Integrating Functions of Several Variables
 The definite Integral

Iterated Integrals
 Triple Integrals
 Double Integrals in Polar Coordinates
 Cylindrical and Spherical Coordinates
 Applications to Probability
 Change of Variables
 Chapter 17 Chapter 17: Vector Fields
 Chapter 18 Chapter 18: Line Integrals
 Chapter 19 Chapter 19: Flux Integrals
 Chapter 20 Chapter 20: Calculus of Vector Fields
Section 16.2. Iterated Integrals
Example 1.
Example 2.
Example 3. The density at the point \( (x, y) \)
of a trianglar metal plate, as shown in Figure, is \( \delta (x, y) . \) Explress its mass
as an iterated integral.
First, we plot triangles using Mathematica:
Plot[{0., Max[0, Min[2 x, 6  x]]}, {x, 1, 7}, AspectRatio > 1/2,
Ticks > {{0, 2, 4, 6}, {0, 2, 4}}, Filling > Axis,
FillingStyle > Blue]
Graphics[Polygon[{{1, 0}, {0, Sqrt[3]}, {1, 0}}]]
RegionPlot[2 x  y > 0 && 3 x  y > 15, {x, 0, 5}, {y, 0, 6},
PlotStyle > LightBlue]
Plot[{0, Max[0, Min[2 x, 15  3 x]]}, {x, 0, 7}, AxesLabel > {x, y},
Filling > Bottom]
RegionPlot[
2 x  y > 1 && 2 x  y > 5 && y > 1, {x, 0, 3}, {y, 0, 3},
PlotStyle > Purple]
RegionPlot[1 < x < 3 && 2 < y < 1 && 3/4 x  y > .25, {x, 2,
5}, {y, 3, 3}, GridLines > Automatic]
RegionPlot[x < 4 && y > 2 && 2 x  y > 0, {x, 0, 6}, {y, 0, 8}]
Show[Graphics[{PlotStyle > {Thickness[1.5]}, Hue[0.8],
Line[{{0, 0}, {1, 1}, {4, 1/2}, {0, 0}}]}], AspectRatio > 1,
Axes > True, AxesStyle > Thick, AxesOrigin > {0.5, 0.5}]
RegionPlot[x < 1 && 2 x  y > 0, {x, 0, 1}, {y, 0, 2},
PlotStyle > Hue[0.2]]
RegionPlot[x > 0 && 3 x  y < 0 && x < 12, {x, 0, 6}, {y, 0, 12}]
R = Triangle[{{1, 1}, {1, 12}, {4, 12}}]
Graphics[{EdgeForm[Dashed], Pink, R}, AspectRatio > 0.4,
Axes > True, AxesStyle > Thick, AxesOrigin > {0, 0}]
If the boundary needs to be emphasized, then
ln = Show[
Graphics[{Thickness[.02], Hue[.6],
Line[{{0, 0}, {1, 1}, {0, 1}, {0, 0}}]}]]
pol = Show[Graphics[{Hue[0.5], Polygon[{{0, 0}, {1, 1}, {0, 1}}]}]]
Show[pol, ln]
An arbitrary triangle could be plotted as follows
triangle[a_?NumericQ, b_?NumericQ, c_?NumericQ] :=
Block[{x, y, pt, sqr}, sqr = #.# &;
pt[a1_, b1_, c1_] :=
Reduce[sqr[{x, y}] == b1^2 && sqr[{x, y}  {a1, 0}] == c1^2 &&
y > 0, {x, y}];
{(Polygon[{{0, 0}, {a, 0}, {x, y}}]),
Text[Style[Framed[a, Background > LightYellow], 11], {a/2, 0}],
Text[Style[Framed[b, Background > LightYellow], 11], {x/2, y/2}],
Text[Style[Framed[c, Background > LightYellow], 11], {(a + x)/2,
y/2}]} /. ToRules[pt[a, b, c]]]
g[{s1_, s2_, s3_}] :=
Graphics[{EdgeForm[Thick], FaceForm[None], triangle[s1, s2, s3]},
ImagePadding > 20, ImageSize > {200, 200}]
GraphicsGrid[{{g[{Sqrt[5], 1, 2}]}}]
or
Graphics[{EdgeForm[Thick], FaceForm[None], triangle[Sqrt[5], 1, 2]}]
Example 4.
Find the mass M of a metal plate R bounded by \( y=x \quad\mbox{and} \quad y= x^2 , \)
with density given by \( \delta (x,y) = 1 + xy \) kg/meter^{2} .
Plot[{x, x^2 }, {x, 0, 1.0}, Filling > {1 >{2}}, AspectRatio >1]
Therefore, we need tools to plot a domain below or above a graph:
q[p_] = 10.2 p^2.0;
lowerBound = p /. Solve[q[p] == 100, p][[1]] // Quiet;
Show[Plot[q[p], {p, 0, 22}, PlotRange > {0, Automatic}],
Plot[q[p], {p, lowerBound, 20}, Filling > 100],
Graphics[{Line[{{0, 100}, {30, 100}}], Line[{{20, 0}, {20, 4000}}]}]]
boundleft = 2;
boundright = 8;
f[x_] = 25  15*x^2 + x^3;
g1 = Plot[f[x], {x, 2, 10}];
g2 = Plot[f[x], {x, boundleft, boundright}, Filling > 0];
Show[g1, g2]
or
Plot[{25  15*x^2 + x^3, If[2 <= x <= 8, 0]}, {x, 0, 10},
PlotStyle > {Automatic, None}, Filling > {1 > {2}}]
or we can fill bellow and above some horizontal line:
guess = 100;
Plot[{25  15*x^2 + x^3, If[2 <= x <= 8, 0]}, {x, 0, 10},
PlotStyle > {Automatic, None}, Filling > {1 > {guess, Yellow}}]
Plot[{25  15*x^2 + x^3, If[2 <= x <= 8, 0]}, {x, 0, 10},
PlotStyle > {Automatic, None}, Filling > {1 > Top}]
Plot[{25  15*x^2 + x^3, If[2 <= x <= 8, 0]}, {x, 0, 10},
PlotStyle > {Automatic, None},
Filling > {1 > {Bottom, LightGreen}}]
Example 5.
 