Given the unit ball: $$M:=\mathbb{B}:\quad\partial M=\varnothing$$
Consider the top-degree form: $$\omega:=1\mathrm{d}x\wedge\mathrm{d}y=\mathrm{d}(x\mathrm{d}y)=:\mathrm{d}\Omega$$
Then one has by Stoke's theorem: $$\pi=\int_\mathbb{B}1=\int_\varnothing\Omega=0$$ But where is the flaw??
The problem is that the form is not compactly-supported.
Also as soon as one introduces a positive bump it fails to have a primitive.