Need to calculate the following triple integral: $\cos(x+2y+3z)$ over a region $D$, where $D=\{(x,y,z): x^2+y^2+z^2 \le1\}$.
The solid $D$ suggests that I should try using spherical coordinates but then I am not able to simplify the function itself.
This is a question from one of my second year multivariable calculus assignments.
So far I have tried using spherical coordinates and using the trigonometric identities.
Any other approach that I could try?
As explained by user90189,
$$\int_D\cos(x+2y+3z)\,dx\,dy\,dz=\int_D\cos(\sqrt{14}z)\,dx\,dy\,dz.$$
In cylindrical coordinates,
$$2\pi\int_{r=0}^1\int_{z=-\sqrt{1-r^2}}^{\sqrt{1-r^2}}\cos(\sqrt{14}z)\,dz\,r\,dr.$$