222 - Topic 17 - Volumes over irregular domains

574 days ago by Professor222

Graphing non-rectangular domain surfaces for multivariate Calculus

John Travis

Mississippi College

In the worksheet below, volumes for various solids are graphed when the solid is the region below a given surface.  The user must enter inequalities describing the region (in x and y) over which the surface is of interest. 

Rectangular viewing boundaries should be adjusted using the sliding x and y range values.

The color of the top surface can be changed using the color selector box.  An intermediate surface can be determined by clicking on the shade volume checkbox.

## Graphing surfaces over non-rectangular domains ## John Travis ## Spring 2011 ## ## Interact allows the user to input up to two inequality constraints on the ## domain when dealing with functional surfaces ## ## User inputs: ## f = "top" surface with z = f(x,y) ## g = "bottom" surface with z = g(x,y) ## condition1 = a single boundary constraint. It should not include && or | to join two conditions. ## condition2 = another boundary constraint. If there is only one constraint, just enter something true ## or even just an x (or y) in the entry blank. ## ## var('x,y') # f is the top surface # g is the bottom surface global f,g # condition1 and condition2 are the inequality constraints. It would be nice # to have any number of conditions connected by $$ or | global condition1,condition2 @interact(layout=dict(top=[['f','condition1'],['g','condition2']],right=[['show_3d'],['dospin'],['clr'],['auto_update']],bottom=[['xx'],['yy']])) def _(f=input_box(default=(1/3)*x^2 + (1/4)*y^2 + 5,width=30,label='$f(x)=$'), g=input_box(default=-1*x+0*y,width=30,label='$g(x)=$'), condition1=input_box(default= x^2+y^2<8,width=30,label='. Constraint$_1=$'), condition2=input_box(default=y<sin(3*x),width=30,label='. Constraint$_2=$'), show_3d=('Stereographic',false), show_vol=('Shade volume',true), dospin = ('Spin?',true), clr = color_selector('#faff00', label='Volume Color', widget='colorpicker', hide_box=True), xx = range_slider(-5, 5, 1, default=(-3,3), label='X Range'), yy = range_slider(-5, 5, 1, default=(-3,3), label='Y Range'), auto_update=false): # This is the top function actually graphed by using NaN outside domain def F(x,y): if condition1(x=x,y=y): if condition2(x=x,y=y): return f(x=x,y=y) else: return -NaN else: return -NaN # This is the bottom function actually graphed by using NaN outside domain def G(x,y): if condition1(x=x,y=y): if condition2(x=x,y=y): return g(x=x,y=y) else: return -NaN else: return -NaN # This interpolates the "volume" between F and G. Replaced with an inline version # in the "loopish thing" below. ## def H(x,y): ## if condition1(x=x,y=y): ## if condition2(x=x,y=y): ## return (1-r)*f(x=x,y=y)+r*g(x=x,y=y) ## else: ## return -NaN ## else: ## return -NaN ## P = Graphics() # The graph of the top and bottom surfaces P_list = [] P_list.append(plot3d(F,(x,xx[0],xx[1]),(y,yy[0],yy[1]),color='blue',opacity=0.9)) P_list.append(plot3d(G,(x,xx[0],xx[1]),(y,yy[0],yy[1]),color='gray',opacity=0.9)) # Interpolate "layers" between the top and bottom if desired if show_vol: ratios = range(10) def H(x,y,r): return (1-r)*F(x=x,y=y)+r*G(x=x,y=y) P_list.extend([ plot3d(lambda x,y: H(x,y,ratios[1]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[2]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[3]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[4]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[5]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[6]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[7]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[8]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr), plot3d(lambda x,y: H(x,y,ratios[9]/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.2,color=clr) ]) # P = plot3d(lambda x,y: H(x,y,ratio/10),(x,xx[0],xx[1]),(y,yy[0],yy[1]),opacity=0.1) # Now, accumulate all of the graphs into one grouped graph. P = sum(P_list[i] for i in range(len(P_list))) if show_3d: show(P,frame=true,axes=false,xmin=xx[0],xmax=xx[1],ymin=yy[0],ymax=yy[1],stereo='redcyan',figsize=(4,6),viewer='jmol',spin=dospin) else: show(P,frame=true,axes=false,xmin=xx[0],xmax=xx[1],ymin=yy[0],ymax=yy[1],figsize=(4,6),viewer='jmol',spin=dospin) 
       

Click to the left again to hide and once more to show the dynamic interactive window

# Trying out Larson "Calculus with Early Transcendentals", 5th ed., #50, page 1002 var('x,y,z') assume(x>0,y>0,4-sqrt(y)>0) f = 1 I1 = integrate(f,z,0,ln(1+x+y)) show(I1) I2 = integrate(I1,x,0,4-sqrt(y)) show(I2) I3 = integrate(I2,y,0,16) show(I3) 
       

                                
                            

                                
var('x,y,z') assume(x>0,y>0,x-4<0) f = 1 I1 = integrate(f,z,0,ln(1+x+y)) show(I1) I2 = integrate(I1,y,0,(x-4)^2) show(I2) I3 = integrate(I2,x,0,4) show(I3)