An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}$. Find the total charge on $R$ if the charge density at a point $( x , y )$ in $R$ (measured in coulombs per square meter) is $\sigma ( x , y ) = 4 x ^ { 2 } + y ^ { 3 }$.
A) 84 coulombs
B) 222 coulombs
C) 22 coulombs
D) 56 coulombs

Question

Quizplus · Accepted Answer

To find the total charge on $R$, integrate the charge density $\sigma(x, y) = 4x^2 + y^3$ over the region $R$. The total charge $Q$ is given by the double integral $\int_{0}^{3} \int_{0}^{2} (4x^2 + y^3) dy dx$. First, integrate with respect to $y$, treating $x$ as a constant, and then integrate the result with respect to $x$. The calculation yields $Q = 84$ coulombs.

An Electric Charge Is Spread Over a Rectangular Region R={(x,y)∣0≤x≤3,0≤y≤2}R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}R={(x,y)∣0≤x≤3,0≤y≤2}

An Electric Charge Is Spread Over a Rectangular Region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}$