So consider the smooth map defined by
$$f: \Bbb{R}^2 \rightarrow \Bbb{R}^3$$
via
$$(\theta, \phi) \to ((2 + \cos \phi)\cos \theta,(2 +\cos \phi)\sin \theta, \sin \phi)$$
Let $M:=F(\Bbb{R}^2)$ as the $2$ torus with the orientation that makes $f$ orientation preserving. Then compute $f^*\omega$ where $\omega$ Is the $2$-form
$$\omega = x \, dy \, \wedge \, dz$$
I computed $f^*\omega$ here to be
$$f^*\omega=(2+\cos \phi)^2\cos^2 \theta \cos \phi \, d \theta \, \wedge d \phi$$
I need to use this to compute $\int_M \omega$ which by diffeomorphism invariance, this just equals
$$\int_{\Bbb{R}^2} f^* \omega = \int_{\Bbb{R}^2}(2+\cos \phi)^2\cos^2 \theta \cos \phi \, d \theta \, \wedge d \phi$$
(Here $$f:\Bbb{R}^2 \rightarrow M$$ is the function I'm applying diffeomorphism invariance to) which is a double integral
$$\int_{-\infty}^\infty \int_{-\infty}^\infty (2+\cos \phi)^2\cos^2 \theta \cos \phi \, d \theta \, d \phi$$
Am I correct thus far? In computing $\int_M \omega$.