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## Homework Statement

find the surface area of a sphere shifted R in the z direction using spherical coordinate system.

## Homework Equations

$$S= \int\int \rho^2 sin(\theta) d\theta d\phi$$

$$x^2+y^2+(z-R)^2=R^2$$

## The Attempt at a Solution

I tried to use the sphere equation mentioned above and solve for ρ, using the definition of ρ and z in spherical coordinate system, this gives me:

$$x^2+y^2+z^2+R^2-2 R z= R^2$$

$$\rho^2-2 R \rho cos(\theta)= 0$$

$$\rho= 2 R cos(\theta)$$

then substitute in the integral:

$$S= \int\int 4 R^2 cos(\theta)^2 sin(\theta) d\theta d\phi$$

and integrate with the following boundaries:

θ: 0 → π/2

Φ: 0 → 2π

this gives a wrong answer of:

$$\frac{8 \pi R^2}{3}$$

What am I missing? This is a very basic problem but I cannot put my hand on the mistake.