Electric charge is distributed over the rectangle 2 ≤ x ≤ 4, 0 ≤ y ≤ 2 so that the charge density at (x, y) is σ(x, y) = 2xy + y2 (measured in coulombs per square meter). Find the total charge on the rectangle.

The total charge on the rectangle is 29.33C

Step-by-step explanation:

Given

σ(x, y) = 2xy + y²

For 2 ≤ x ≤ 4, 0 ≤ y ≤ 2

The total charge on the rectangle is calculated as follows.

Charge, q = ∫∫2xy + y² dydx {0,2}{2,4}

Integrate with respect to y

q = ∫xy² + ⅓y³ dydx {0,2}{2,4}

q = ∫(2²*x + ⅓*2³) dx {2,4}

q = ∫4x + 8/3 dx {2,4}

Integrate with respect to x

q = 2x² + 8x/3 {2,4}

q = (2(4)² + 8(4)/3) - (2(2)² + 8(2)/3)

q = 29.33C