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- Is it possible to integrate $$\frac {1} {2} \int_0^\pi (sin(\theta)\uparrow \uparrow \infty)^{2} d\theta$$

I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs both do in fact seem to be converging to a definite number. I know this is all kind of messy so I'm refining my question to make things clearer. I am wondering if it is possible to evaluate the following intergral:

$$\frac {1} {2} \int_0^\pi (sin(\theta)\uparrow \uparrow \infty)^{2} d\theta$$

$$\frac {1} {2} \int_0^\pi (sin(\theta)\uparrow \uparrow \infty)^{2} d\theta$$