Defference between $\delta(|\boldsymbol{r}| - 1)$ and $\chi_{S^{2}}(\boldsymbol{r})$ in three-dimensional integration

Question

Are the following equations incorrect? $$ \int_{\mathbb{R}^{3}} f(\boldsymbol{r})\delta(|\boldsymbol{r}| - 1) \mathrm{d}\boldsymbol{r} = \int_{S^{2}} f(\boldsymbol{r}) \mathrm{d}\boldsymbol{r} \tag{1}, $$ $$ \int_{\mathbb{R}^{3}} f(\boldsymbol{r})\chi_{S^{2}}(\boldsymbol{r}) \mathrm{d}\boldsymbol{r} = \int_{S^{2}} f(\boldsymbol{r}) \mathrm{d}\boldsymbol{r} \tag{2}, $$ where $\delta$ is Dirac's delta and $\chi_{\Omega}$ is characteristic function with support $\Omega$. If these are correct, it makes me wonder.