2.05.2DIntegrals

# 2.05.2DIntegrals - You can plot a surface z = [email protected] yD like...

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Accumulation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 2.05 2D Integrals and the Gauss-Green Formula BASICS B.1) 2D integrals Ÿ Ÿ R f @ x, y D x y for volume measurements Here's a plot of a function y = f @ x D : Clear @ f, x D ; f @ x _ D = E - 0.2 x Cos @ 4 x D ; 8 a, b < = 8 0, 4 < ; fplot = Plot @ f @ x D , 8 x, a, b < , PlotStyle Ø 88 Blue, Thickness @ 0.01 D<< , AxesLabel Ø 8 "x" , "y" <D 1 2 3 4 x - 0.5 0.5 1.0 y To get the points on the plot, you go to a point 8 x 0 , 0 < on the x-axis underneath or above the plot and then you run a line to the point 8 x 0 , f @ x 0 D< on the curve like this: xo = 1.5; Show @ fplot, Graphics @8 PointSize @ 0.02 D , Point @8 xo, 0 <D<D , Graphics @8 PointSize @ 0.02 D , Point @8 xo, f @ xo D<D<D , Graphics @8 Red, Line @88 xo, 0 < , 8 xo, f @ xo D<<D<DD 1 2 3 4 x - 0.5 0.5 1.0 y Play with this by resetting x 0 and rerunning. You can get a pretty good idea of what the curve looks like by just looking at the tips of the little line segments: xjump = b - a 50 ; Show @ fplot, Table @8 Graphics @8 PointSize @ 0.02 D , Point @8 x, 0 <D<D , Graphics @8 PointSize @ 0.02 D , Point @8 x, f @ x D<D<D , Graphics @8 Red, Line @88 x, 0 < , 8 x, f @ x D<<D<D< , 8 x, a, b, xjump <DD 1 2 3 4 x - 0.5 0.5 1.0 y You can plot a surface z = f @ x, y D like this: In[1]:= Clear @ f, x, y D ; f @ x _ , y _ D = 3.1 x 2 + 2.3 y 2 ; 88 a, b < , 8 c, d << = 88 - 2, 3 < , 8 - 1, 4 << ; surfaceplot = Plot3D @ f @ x, y D , 8 x, a, b < , 8 y, c, d <D ; spacer = 0.2; threedims = Graphics3D @8 8 Blue, Line @88 a, 0, 0 < , 8 b, 0, 0 <<D< , Text @ "x" , 8 b + spacer, 0, 0 <D , 8 Blue, Line @88 0, c, 0 < , 8 0, d, 0 <<D< , Text @ "y" , 8 0, d + spacer, 0 <D , 8 Blue, Line @88 0, 0, 0 < , 8 0, 0, 50 <<D< , Text @ "z" , 8 0, 0, 50 + spacer <D<D ; CMView = 8 2.7, 1.6, 1.2 < ; fplot = Show @ surfaceplot, threedims, ViewPoint -> CMView, PlotRange -> All D Out[8]= · B.1.a) Explain the meaning of the plotted points that make up the surface. · Answer: You do it just as you did it above. To get the points on the plot, you go to a point 8 x 0 , y 0 , 0 < on the xy-plane underneath or above the surface. Then you run a line to the 3 D point 8 x 0 , y 0 , f @ x 0 , y 0 D< like this: In[9]:= 8 xo, yo < = 8 2.5, 3 < ; Show @ fplot, Graphics3D @8 PointSize @ 0.02 D , Point @8 xo, yo, 0 <D<D , Graphics3D @8 PointSize @ 0.02 D , Point @8 xo, yo, f @ xo, yo D<D<D , Graphics3D @8 Red, Line @88 xo, yo, 0 < , 8 xo, yo, f @ xo, yo D<<D<DD Out[10]= Play with this by resetting x 0 and y 0 and rerunning. You can get a pretty good idea of what the surface looks like by just looking at the tips of the little line segments: In[11]:= xjump = b - a 8 ; yjump = d - c 8 ; Show @ fplot, Table @8 Graphics3D @8 PointSize @ 0.02 D , Point @8 x, y, 0 <D<D , Graphics3D @8 PointSize @ 0.02 D , Point @8 x, y, f @ x, y D<D<D , Graphics3D @8 Red, Line @88 x, y, 0 < , 8 x, y, f @ x, y D<<D<D< , 8 x, a, b, xjump < , 8 y, c, d, yjump <DD

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2.05.2DIntegrals - You can plot a surface z = [email protected] yD like...

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