Prof. Vladimir Dobrushkin
Department of Mathematics
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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.

 

Section 16.7. Change of Variables

Crescent Moon:
Show[ContourPlot[{x^2 + y^2 == 16, x^2 + y^2/4 == 4}, {x, 0, 4}, {y, -4, 4}, AspectRatio -> 1.4], RegionPlot[{x^2 + y^2 == 16, x^2 + y^2/4 == 4}, {x, 0, 4}, {y, -4, 4}, AspectRatio -> 1.4], RegionPlot[x^2 + y^2/4 >= 4 && x^2 + y^2 < 16, {x, 0, 4}, {y, -4, 4}, AspectRatio -> 1.4]]
or
figure
 th = linspace( pi/2, -pi/2, 100);
R = 1;  
x = R*cos(th) + 5;
y = R*sin(th) + 4;
plot(x,y); axis equal;
title('figure 16.22')
Plot of crescent without shading
Show[PolarPlot[{8 Cos[t]}, {t, 0, 10}], PolarPlot[6, {t, 0, 10}], Axes -> Automatic, PlotRange -> {{4, 8}, {-4, 4}}]
ContourPlot[{x^2 + y^2 == 16, x^2 + y^2/4 == 4}, {x, 0, 4}, {y, -4, 4}, AspectRatio -> 1.5, PlotTheme -> "Scientific", ColorFunction -> "Rainbow" ]
RegionPlot[ x^2 + y^2 <= 16 && (x + 2)^2 + y^2 >= 20, {x, -5, 5}, {y, -5, 5}]
Show[Graphics[{RGBColor[0,1,0], Circle[{2, 1}, 1]]}, PlotRangePadding ->0.8], RegionPlot[{(x - 2)^2 + (y - 1)^2 < 1 && (x - 1.5)^2 + (y - 1)^2 > 1}, {x, -3, 3}, {y, -3, 6}, PlotPoints -> 200, PlotRange -> {0, 3}]]
With[{a = 1, b = 1.212, c = -0.7}, Show[{ RegionPlot[(x^2 + y^2 < a^2 && (x - c)^2 + y^2 > b^2), {x, -1.2, 2}, {y, -1.2, 1.2}, AspectRatio -> Automatic, Frame -> False, PlotStyle -> Hue[.5], MaxRecursion -> 4], Graphics[{Circle[{0, 0}, a], Circle[{c, 0}, b] (*,PointSize[.015],Point[{{0,0},{c,0}}]*) }] }, Method -> {Antialiasing -> True}]]
or flipping
With[{a = 1, b = 1.212, c = .7}, Show[{ RegionPlot[(x^2 + y^2 < a^2 && (x - c)^2 + y^2 > b^2), {x, -1.2, 2}, {y, -1.2, 1.2}, AspectRatio -> Automatic, Frame -> False, PlotStyle -> Hue[.5], MaxRecursion -> 4], Graphics[{Circle[{0, 0}, a], Circle[{c, 0}, b] (*,PointSize[.015],Point[{{0,0},{c,0}}]*) }] }, Method -> {Antialiasing -> True}]]