restart;with(plots):with(Student[Calculus1]): with(plottools): f :=x->exp(-x); plot(f,0..4); The following comamands can be used to create an animation that shows the curve revolving around the x-axis start := spacecurve([0,x,f(x)],x=0..4,thickness=3): pic := n-> cylinderplot(f(z),theta=0..n*2*Pi/30,z=0..3): display(start,seq(rotate(pic(n),Pi/2,Pi/2,0),n=1..30), insequence=true,axes=normal,tickmarks=[0,0,0]); The following command will show the 3D plot. VolumeOfRevolution(exp(-x), 0..4, 'axis'='horizontal', 'output'='plot'); The following commands show how we can derive the formula for the volume. p1:=plot(f(x),x=0..4): p2:=plot([[4,0],[4,f(4)]]): e1:=polygonplot([[1,0],[1.2,0],[1.2,f(1.2)],[1,f(1)]],color=green): t1:=textplot([1.1,.39,convert([68],bytes)],font=[SYMBOL,12]): t2:=textplot([1.2,.39,`x`]): display({p1,p2,e1,t1,t2}); The following command is an alternative way of creating the 3D plot. radius:='radius':with(plots): p4:=tubeplot([x,0,0],x=0..4,radius=f(x),axes=normal): display(p4,orientation=[-60,75]); Calculate the volume exactly, as follows: v:=Pi*Int(f(x)^2,x=0..4); v1:=value(v); evalf(v1); VolumeOfRevolution(exp(-x), 0..4, 'axis'='horizontal', 'output'='value'); evalf(%);

Now we can have the intersection of 2 curves revolved around the x-axis. p1:= tubeplot([x,0,0],x=0..1,radius=x): p2:=tubeplot([x,0,0],x=0..1,radius=x^2): display({p1,p2},axes=normal,orientation=[-60,75],style=patch); VolumeOfRevolution(x,x^2, 0..1, 'axis'='horizontal', 'distancefromaxis'=0,'output'='plot'); VolumeOfRevolution(x,x^2, 0..1, 'axis'='horizontal', 'distancefromaxis'=0,'output'='value'); evalf(%); VolumeOfRevolution(exp(-x), 0..4, 'axis'='horizontal', 'distancefromaxis'=-1,'output'='plot'); 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