Consider the following integral, $$1/(4\pi^2) \int_0^{2\pi} \int_0^{2\pi} (9- \sin^2 \frac{\theta_1 }{2} \sin^2 \frac{\theta_2 }{2})^{1/2} d\theta_1 d\theta_2.$$ This integral comes up in computing the volume of 3 dimensional special orthogonal matrices of Hessenberg form, i.e., the bottom left entry is $0$. Mathematica isn't able to produce close form solution. Numerically it's about 2.95.

Consider the following integral, $$ {1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi} \sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right) \sin^{2}\left(\theta_{2} \over 2\right)\,} \,{\rm d}\theta_{1}\,d\theta_{2} $$ This integral comes up in computing the volume of $3$-dimensional special orthogonal matrices of Hessenberg form, i.e., the bottom left entry is $0$. Mathematica isn't able to produce close form solution. Numerically it's about $2.95$.

Consider the following integral, $$ \int_0^{2\pi} \int_0^{2\pi} (9- \sin^2 \frac{\theta_1 }{2} \sin^2 \frac{\theta_2 }{2})^{1/2} d\theta_1 d\theta_2.$$ This integral comes up in computing the volume of 3 dimensional special orthogonal matrices of Hessenberg form, i.e., the bottom left entry is $0$. Mathematica isn't able to produce close form solution. Numerically it's about 2.95.

Consider the following integral, $$1/(4\pi^2) \int_0^{2\pi} \int_0^{2\pi} (9- \sin^2 \frac{\theta_1 }{2} \sin^2 \frac{\theta_2 }{2})^{1/2} d\theta_1 d\theta_2.$$ This integral comes up in computing the volume of 3 dimensional special orthogonal matrices of Hessenberg form, i.e., the bottom left entry is $0$. Mathematica isn't able to produce close form solution. Numerically it's about 2.95.