Department of Mathematics

 MTH243 (Calculus for Functions of Several Variables) SAGE. Chapter 16: Integrating Functions of Several Variables Vladimir A. Dobrushkin,Lippitt Hall 202C, 874-5095,dobrush@uri.edu In this course we will use Sage computer algebra system (CAS), which is a free software. The Sage projects are created to help you learn new concepts. Sage 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. The university has a license for computer algebra system Mathematica, so it is free to use for its students. A student can also use free CASs: SymPy (based on Python), or Maxima.
• 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/meter2 .

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.