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

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