The rectangular region of the xy plane shown has nonuniform surface charge density σ = σ0 (1+ y b), where σ0 is a constant. find the net charge.

This is net charge on the surface  is  Q = σ₀ x (y + 2by²)

Explanation:

The surface charge density is defined as the amount of charge Q per unit area A

       σ = dq / dA

       dq = σ dA

Since the surface is a rectangular region we use an xy coordinate system so the area difference  

      dA = dxdy

      dq = σ dx dy

 We replace, evaluate the integral

        ∫ dq = ∫ σ₀ (1 + yb) dxdy

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        Q -0 =σ₀ ∫ (1 + yb) (x-0)   dy

Where we evaluate We must recognize that the charge Q must be zero by the time X = 0 and Y = 0. At the starting point Q = 0 for x = 0

We perform the other integral (dy)

        Q = σ₀ x (y + 2y² b)

Evaluated between Y = 0 and Y = y

      Q = σ₀ x (y + 2by²)

This is net charge on the surface

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