Answer
The total charge is $285$ coulombs.
Work Step by Step
Given:
- A rectangle is $0\leq x\leq 5$ and $2\leq y\leq 5$
- Electric charge density at $(x,y)$ is $\sigma(x,y)=2x+4y$
The total charge is given by $T=\int_0^5 \int_2^5 \sigma (x,y) dydx$:
Evaluate $T$:
$T=\int_0^5\int_2^52x+4ydydx$
$=\int_0^5 2xy+2y^2]_2^5 dx$
$=\int_0^5(10x+50)-(4x+8)dx$
$=\int_0^5 6x+42dx$
$=3x^2+42x]_0^5$
$=75+210-0$
$=285$
