Finding the determinant of a $3*3$ matrix that was developed using LaGrangian multiplier to optimize the surface area of a conical frustum

Question

This is the matrix I wrote down. I'm struggling to find its determinant as by hand it would be impractical and would take a lot of time, is there any online calculator or a way to find it quickly, if so how, thanks.

Answer 1

Using Symbolab, I got:

$$\det \begin{pmatrix}\pi \left(1+sin\left(\theta \right)\right)&\pi \left(1+sin\left(\theta \right)\right)&\frac{1}{6}\pi \:\:cos\left(\theta \right)\left(ysin\left(\theta \right)+3x\right)\\ \:\:\:\pi \:&0&\frac{1}{6}\pi \:\:cos\left(\theta \right)\left(3x-ysin\left(\theta \right)\right)\\ \:\:\:0&\pi \:\:cos\left(\theta \right)&\frac{1}{24}\pi \left(6x^2+y^2-y^2cos\left(2\theta \right)\right)\end{pmatrix}$$

$$=-\frac{\pi ^3x\cos ^2\left(θ\right)\sin \left(θ\right)}{2}+\frac{2\pi ^3y\cos ^2\left(θ\right)\sin \left(θ\right)+\pi ^3y\cos ^2\left(θ\right)\sin ^2\left(θ\right)}{6}+\frac{-\pi ^3y^2+\pi ^3y^2\cos \left(2θ\right)-\pi ^3y^2\sin \left(θ\right)+\pi ^3y^2\sin \left(θ\right)\cos \left(2θ\right)}{24}+\frac{-\pi ^3x^2-\pi ^3x^2\sin \left(θ\right)}{4}$$