# Evaluate the following. (x-5y+z) ds S S S: z = 13 - 8x + 8y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 3

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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

Therefore:

`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

`int_0^1`-96(3-0) dx

**Sources:**

`int_s ` `int` (x-5y+z) ds

*S*: *z* = 13 - 8*x* + 8*y*, 0 ≤ *x* ≤ 1, 0 ≤ *y* ≤ 3