Denote the area of the sphere $B_R$ of radius $R$ by $A_R$, and denote by $r$ the radial coordinate, then $$A_R=\int_{B_R}\delta(r-R)\mu(\mathbf{x})\,d\mathbf{x},$$ with $\delta(r)$ the Dirac delta function.

Example in three dimensions, with $\mu(\mathbf{x})\,d\mathbf{x}=r^2 \sin\theta\, drd\theta d\phi$ in spherical coordinates,

$$A_R=\int_0^R \delta(r-R) r^2 dr \int_0^{2\pi} d\phi \int_0^\pi \sin \theta d\theta =4\pi R^2,$$ where I have used that $\int f(s)\delta(s-u)ds=f(u)$.

Denote by $A_R$ the area of the ball $B_R$ in $\mathbb{R}^d$ of radius $R$, and denote by $r$ the radial coordinate, then $$A_R=\int_{\mathbb{R}^{d}}\delta(r-R)\mu(\mathbf{x})\,d\mathbf{x},$$ with $\delta(r)$ the Dirac delta function.

Example in three dimensions, $d=3$, with measure $\mu(\mathbf{x})\,d\mathbf{x}=r^2 \sin\theta\, drd\theta d\phi$ in spherical coordinates,

$$A_R=\int_0^\infty \delta(r-R) r^2 dr \int_0^{2\pi} d\phi \int_0^\pi \sin \theta d\theta =4\pi R^2.$$

Denote the area of the sphere $B_R$ of radius $R$ by $A_R$, and denote by $r$ the radial coordinate, then $$A_R=\int_{B_R}\delta(r-R)\mu(\mathbf{x})\,d\mathbf{x},$$ with $\delta(r)$ the Dirac delta function.

Example in three dimensions, with $\mu(\mathbf{x})\,d\mathbf{x}=r^2 sin\theta drd\theta d\phi$,

$$A_R=\int_0^R \delta(r-R) r^2 dr \int_0^{2\pi} d\phi \int_0^\pi \sin\theta d\theta =4\pi R^2,$$ where I have used that $\int f(x)\delta(x-u)dx=f(u)$.

Denote the area of the sphere $B_R$ of radius $R$ by $A_R$, and denote by $r$ the radial coordinate, then $$A_R=\int_{B_R}\delta(r-R)\mu(\mathbf{x})\,d\mathbf{x},$$ with $\delta(r)$ the Dirac delta function.

Example in three dimensions, with $\mu(\mathbf{x})\,d\mathbf{x}=r^2 \sin\theta\, drd\theta d\phi$ in spherical coordinates,

$$A_R=\int_0^R \delta(r-R) r^2 dr \int_0^{2\pi} d\phi \int_0^\pi \sin \theta d\theta =4\pi R^2,$$ where I have used that $\int f(s)\delta(s-u)ds=f(u)$.