︠7d027468-a562-435a-81c6-c8efe4d4e4a9i︠
%html
<p><span style="font-size: large;">Computing the line integral and viewing the lateral surface</span></p>
<p><span style="font-size: large;">John Travis</span></p>
<p><span style="font-size: large;">Mississippi College</span></p>
<p><span style="font-size: large;">Tags:&nbsp;</span></p>
<ul>
<li><span style="font-size: large;">Multivariate Calculus</span></li>
<li><span style="font-size: large;">Lateral Surface</span></li>
<li><span style="font-size: large;">Line Integral</span></li>
</ul>

︡02acffa1-fcd8-47f4-9b5e-6d76bff28fed︡{"html": "<p><span style=\"font-size: large;\">Computing the line integral and viewing the lateral surface</span></p>\n<p><span style=\"font-size: large;\">John Travis</span></p>\n<p><span style=\"font-size: large;\">Mississippi College</span></p>\n<p><span style=\"font-size: large;\">Tags:&nbsp;</span></p>\n<ul>\n<li><span style=\"font-size: large;\">Multivariate Calculus</span></li>\n<li><span style=\"font-size: large;\">Lateral Surface</span></li>\n<li><span style=\"font-size: large;\">Line Integral</span></li>\n</ul>"}︡
︠b4c7ce1c-36c3-48e5-8230-31b85512c55bai︠
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##  Display and compute the area of the lateral surface between two surfaces
##  corresponding to the (scalar) line integral
##  John Travis
##  Spring 2011

var('x,y,t,s')
@interact
def _(f=input_box(default=6-4*x^2-y^2*2/5,label='$f(x,y) = $'),
        g=input_box(default=-2+sin(x)+sin(y),label='$g(x,y) = $'),
        u=input_box(default=cos(t),label='$u(t) = $'),
        v=input_box(default=2*sin(t),label='$v(t) = $'),
        a=input_box(default=0,label='$a = $'),
        b=input_box(default=3*pi/2,label='$b = $'),
        xx = range_slider(-5, 5, 1, default=(-1,1), label='x view'),
        yy = range_slider(-5, 5, 1, default=(-2,2), label='y view'),
        smoother=checkbox(default=false)):
        
    ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2)
    
#   Set up the integrand to compute the line integral, making all attempts
#   to simplify the result so that it looks as nice as possible.    
    A = (f(x=u(t),y=v(t))-g(x=u(t),y=v(t)))*ds.simplify_trig().simplify()
    
#   It is not expected that Sage can actually perform the line integral calculation.
#   So, the result displayed may not be a numerical value as expected.
#   Creating a good but harder example that "works" is desirable.
    line_integral = integral(A,t,a,b)
    line_integral_approx = numerical_integral(A,a,b)[0]
       
    html(r'<h4 align=center>Lateral Surface Area = $ %s $ </h4>'%latex(line_integral))

    html(r'<h4 align=center>Lateral Surface $ \approx $ %s</h2>'%str(line_integral_approx))

#   Plot the top function z = f(x,y) that is being integrated.
    G = plot3d(f,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2)
    G += plot3d(g,(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2)

#   Add space curves on the surfaces "above" the domain curve (u(t),v(t)) 
    G += parametric_plot3d([u,v,g(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')
    G += parametric_plot3d([u,v,f(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')
    k=0
    if smoother:
        delw = 0.025
        lat_thick = 3
    else:
        delw = 0.10
        lat_thick = 10
    for w in (a,a+delw,..,b):
        G += parametric_plot3d([u(w),v(w),s*f(x=u(w),y=v(w))+(1-s)*g(x=u(w),y=v(w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9)
    show(G,spin=true)
︡f7122216-3386-48e8-8f55-75cc2d26fa4d︡{"html": "<div padding=6 id=\"div-interact-4\"> <table width=800px height=20px bgcolor=\"#c5c5c5\"\n                 cellpadding=15><tr><td bgcolor=\"#f9f9f9\" valign=top align=left><table>\n<tr><td colspan=3><table><tr><td align=right><font color=\"black\"><span class=\"math\">f(x,y) = </span>&nbsp;</font></td><td><input type=\"text\" value=\"-4*x^2 - 2/5*y^2 + 6\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'f\\', 1, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\"><span class=\"math\">g(x,y) = </span>&nbsp;</font></td><td><input type=\"text\" value=\"sin(x) + sin(y) - 2\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'g\\', 2, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\"><span class=\"math\">u(t) = </span>&nbsp;</font></td><td><input type=\"text\" value=\"cos(t)\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'u\\', 3, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\"><span class=\"math\">v(t) = </span>&nbsp;</font></td><td><input type=\"text\" value=\"2*sin(t)\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'v\\', 4, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\"><span class=\"math\">a = </span>&nbsp;</font></td><td><input type=\"text\" value=\"0\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'a\\', 5, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\"><span class=\"math\">b = </span>&nbsp;</font></td><td><input type=\"text\" value=\"3/2*pi\" size=80 onchange=\"interact(4, '_interact_.update(\\'4\\', \\'b\\', 6, _interact_.standard_b64decode(\\''+encode64(this.value)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr><tr><td align=right><font color=\"black\">x view&nbsp;</font></td><td><table><tr><td>\n        <div id=\"slider-xx-4\" style=\"margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;\"></div>\n        </td></tr><tr><td><font color=\"black\" id=\"slider-xx-4-lbl\"></font></td></tr></table><script>(function()\n    {\n        var values = [\"-5\",\"-4\",\"-3\",\"-2\",\"-1\",\"0\",\"1\",\"2\",\"3\",\"4\",\"5\"];\n        var pos = [4, 6];\n        var sel = '#slider-xx-4';\n        var updatePos = function()\n        {\n            pos[0]=$(sel).slider('values', 0);\n            pos[1]=$(sel).slider('values', 1);\n            if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');\n        };\n        setTimeout(function()\n        {\n            $(sel).slider(\n            {\n                range: true,\n                step: 1,\n                min: 0,\n                max: 10,\n                values: [4, 6],\n                change: function(e,ui){ updatePos(); interact(4, '_interact_.update(\\'4\\', \\'xx\\', 7, _interact_.standard_b64decode(\\''+encode64(pos[0]+' '+pos[1])+'\\'), globals()); _interact_.recompute(\\'4\\');'); },\n                slide: updatePos\n            });\n            updatePos();\n        }, 1);\n    })();</script></td>\n</tr><tr><td align=right><font color=\"black\">y view&nbsp;</font></td><td><table><tr><td>\n        <div id=\"slider-yy-4\" style=\"margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;\"></div>\n        </td></tr><tr><td><font color=\"black\" id=\"slider-yy-4-lbl\"></font></td></tr></table><script>(function()\n    {\n        var values = [\"-5\",\"-4\",\"-3\",\"-2\",\"-1\",\"0\",\"1\",\"2\",\"3\",\"4\",\"5\"];\n        var pos = [3, 7];\n        var sel = '#slider-yy-4';\n        var updatePos = function()\n        {\n            pos[0]=$(sel).slider('values', 0);\n            pos[1]=$(sel).slider('values', 1);\n            if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');\n        };\n        setTimeout(function()\n        {\n            $(sel).slider(\n            {\n                range: true,\n                step: 1,\n                min: 0,\n                max: 10,\n                values: [3, 7],\n                change: function(e,ui){ updatePos(); interact(4, '_interact_.update(\\'4\\', \\'yy\\', 8, _interact_.standard_b64decode(\\''+encode64(pos[0]+' '+pos[1])+'\\'), globals()); _interact_.recompute(\\'4\\');'); },\n                slide: updatePos\n            });\n            updatePos();\n        }, 1);\n    })();</script></td>\n</tr><tr><td align=right><font color=\"black\">smoother&nbsp;</font></td><td><input type=\"checkbox\"  width=200px onchange=\"interact(4, '_interact_.update(\\'4\\', \\'smoother\\', 9, _interact_.standard_b64decode(\\''+encode64(this.checked)+'\\'), globals()); _interact_.recompute(\\'4\\');')\"></input></td>\n</tr></table></td></tr>\n<tr><td></td><td style='width: 100%;'><div id=\"cell-interact-4\"><?__SAGE__START>\n        <table border=0 bgcolor=\"white\" width=100%>\n        <tr><td bgcolor=\"white\" align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>\n        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>\n        </table><?__SAGE__END></div></td><td></td></tr>\n<tr><td colspan=3></td></tr>\n</table></td>\n                 </tr></table></div>\n                 "}︡

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