multivariable calculus – approximating unbounded integral in higher dimension

In multivariable calculus, I usually encounter this: if $f: mathbb{R}^n rightarrow mathbb{R}$, we can define $int_{mathbb{R^n}} f(x) dx = lim_{R rightarrow infty} int_{B(0,R)} f(x) dx$ if this limit exists where $B(0,R)$ is the ball centered at $0$ of radius $R$. However, for the case $n=1$, this is not true: consider the function $f(x)=x$. We know that $int_{mathbb{R}} x dx$ is undefined, but $lim_{R rightarrow infty} int_{B(0,R)} x dx = 0$. Can anyone explain to me why?