Evaluate the following. (x-5y+z) ds S S S: z = 13 - 8x + 8y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 3
In order to solve this integral equation you need to use the divergence theorem. In other words, we are calculating the flux of F (x-5y+z) across a surface S.
In order to solve the integral we must first reduce it to the three Cartesian co-ordinates x, y, and z.
Start by taking the derivative of F:
F(x,y,z) = x-5y+z
F'(x,y,z) = 1-5+1 = -3
`int_S` `int` (x-5y+z)ds=`int_x int_y int_z` -3 dxdydz
Next we solve the equation based on the boundary conditions:
zmin = 13-8(0)+8(0)=13
zmax = 13-8(1)+8(3)=45
`int_0^1 int_0^3 int_13^45` -3 dxdydz
`int_0^1 int_0^3` -3(45-13) dxdy