MapleSect88No76.mws

Section 8.8#76

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>    with(plots): 

Warning, the name changecoords has been redefined

>    plot(2*exp(-x),x=0..4,thickness=3);

[Maple Plot]

>    implicitplot3d(y^2+z^2=(2*exp(-x))^2,x=0..4,y=-2..2,z=-2..2,numpoints=100000,axes=boxed,orientation=[245,45]);

[Maple Plot]

>    Int(2*Pi*2*exp(-x)*sqrt(1+4*exp(-2*x)),x=0..infinity);

Int(4*Pi*exp(-x)*(1+4*exp(-2*x))^(1/2),x = 0 .. infinity)

>    value(%);

2*5^(1/2)*Pi+Pi*arctanh(2/5*5^(1/2))

>    evalf(%);

18.58494406

>    Int(2*Pi*(sec(x))^3,x=0..arctan(2));

Int(2*Pi*sec(x)^3,x = 0 .. arctan(2))

>    value(%);

2*5^(1/2)*Pi-1/2*Pi*ln(5)+Pi*ln(5+2*5^(1/2))

>    evalf(%);

18.58494406

Below is the result produced on my web site.

>    Pi*(2*sqrt(5)+ln(2+sqrt(5)));

Pi*(2*5^(1/2)+ln(5^(1/2)+2))

>    evalf(Pi*(2*sqrt(5)+ln(2+sqrt(5))));

18.58494406

>