
MTH243 (Calculus
for Functions of Several Variables)
MAPLE. Chapter 16:
Integrating Functions of Several Variables
Vladimir A. Dobrushkin,Lippitt
Hall 202C, 8745095,dobrush@uri.edu
In this course we will use computer algebra system (CAS) Maple, which is not available in computer
labs at URIit should be purchased separately. The Maple projects are created to help you learn new concepts. It is an essential part of WileyPlus, Matlab, and Mathcad. Maple 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.
We follow the standard textbook "Multivariable Calculus" (6th edition) by McCallum, HughesHallett, Gleason et al.

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}]]
 