Robert Dodier wrote:
> Gary wrote:
>
>> I've been trying to evaluate a symbolic double integral but am
>> perplexed by the unevaluated tan(pi/2) expressions in the result since
>> tan(pi/2) is undefined. What am I missing here and what do I need to
>> do to get this to evaluate to the correct value of
>> (4*pi - 3*sqrt(3))*a^2/6?
>
>> sage input:
>> ***************************************************************
>> var('a r theta')
>> assume(a > 0)
>> integral(integral(r, r, a*csc(theta), 2*a), theta, pi/6, pi/2)
>> ***************************************************************
>>
>> sage output:
>> *********************************************************************
>> (2*pi*tan(pi/2) + 1)*a^2/(2*tan(pi/2)) - (2*pi + 3*sqrt(3))*a^2/6
>> *********************************************************************
>
> Sage punts to Maxima to do symbolic integrals.
> In this case, Maxima computed an antiderivative and plugged
> in the limits of integration. You can coax Maxima into doing
> the right thing by computing a limit:
>
> assume (a > 0, bb > 0, bb > %pi/6);
> integrate (integrate (r, r, a*csc(theta), 2*a), theta, %pi/6, bb);
> => (4*a^2*bb*tan(bb)+a^2)/(2*tan(bb))-(2*%pi+3*sqrt(3))*a^2/6
> limit (%, bb, %pi/2, minus);
> => (4*%pi-3*sqrt(3))*a^2/6
>
> I guess Maxima could apply the limit automatically -- that
> shouldn't change the result for most integrals, but gets the
> correct result in cases like this. Opinions from the Maxima
> crowd are welcome.
Maxima normally does apply limits (or something). There's something
special about the integrand 2*a^2-a^2/2*csc(theta)^2. If you look at
just the integrand csc(theta)^2, maxima returns something nice.
a^2/2*csc(theta)^2 is also ok. Even 2*a^2-a^2*csc(theta)^2 is ok, but
divide csc(theta) term by 2, and maxima is confused.
Don't know why.
Ray