Answer: The answer is 29.33C
Step-by-step explanation:
From the question above, we have the following:
σ(x, y) = 2xy + y²
For 2 ≤ x ≤ 4, 0 ≤ y ≤ 2
The total charge on the rectangle will be calculated as follows.
Charge, q = ∫∫2xy + y² dydx {0,2}{2,4}
Now, we carry out integration with respect to y as follows:
q = ∫xy² + ⅓y³ dydx {0,2}{2,4}
q = ∫(2²*x + ⅓*2³) dx {2,4}
q = ∫4x + 8/3 dx {2,4}
Next, we carry out integration with respect to x as follows:
q = 2x² + 8x/3 {2,4}
q = (2(4)² + 8(4)/3) - (2(2)² + 8(2)/3)
q = (2(16) + 32/3) - (2(4) + 16/3)
q = (32 + 10.67) - (8 + 5.33)
q = 42.67 - 13.33
q = 29.34C
Therefore, the total charge on the rectangle is 29.34C
