An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 4 \} .$ Find the total charge on R if the charge density at a point $( x , y )$ in R (measured in coulombs per square meter) is $\sigma ( x , y ) = x ^ { 2 } + 4 y ^ { 3 }$
A) $804$ coulombs
B) $91$ coulombs
C) $300$ coulombs
D) $265$ coulombs

Question

Quizplus · Accepted Answer

The total charge on R is given by the integral of the charge density over the region R:$$\iint_R \sigma(x,y) dA = \int_0^3 \int_0^4 (x^2 + 4y^3) dy dx = \int_0^3 \left[ x^2 y + y^4 \right]_0^4 dx = \int_0^3 (4x^2 + 256) dx = \left[ \frac{4x^3}{3} + 256x \right]_0^3 = 804$$Therefore, the total charge on R is 804 coulombs.

An Electric Charge Is Spread Over a Rectangular Region R={(x,y)∣0≤x≤3,0≤y≤4}.R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 4 \} .R={(x,y)∣0≤x≤3,0≤y≤4}.

An Electric Charge Is Spread Over a Rectangular Region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 4 \} .$