Evaluate the integral of the function f(x, y, z) = x + y over the surface s given by the following. φ(u, v) = (2u cos(v), 2u sin(v), u), u is in [0, 7], v is in [0, π]

Step-by-step explanation:

Let us recall that if f(x,y,z) is a smooth mapping defined over a surface S, and Φ(u, v) is a smooth parametrization of the surface with (u,v) varying in a certain region R, then

Now,

and

sec ф = 5/3 (quadrant 4)

let x = base

y = perp

z = hypotenuse

sec ф = hypotenuse/base = 5/3

⇒ z = 5 and x = 3

finding y:

z² = x² + y²

5² = 3² + y²

y² = 25-9

y² = 16

y = 4

now finding all other geometric functions of theta:

sin ф = y/z = 4/5

sin ф = -4/5 (sin is negative in the 4th quadrant)

cos ф = x/z = 3/5

cos ф = 3/5 (cos is positive in 4th quadrant)

tan ф = y/x = 4/3

tan ф = -4/3 (tan is negative in 4th quadrant)

cosec ф = z/y = 5/4

cosec ф = -5/4 (cosec is negative in the 4th quadrant)

cot ф = x/y = 3/4

tan ф = -3/4 (tan is negative in 4th quadrant)

step-by-step explanation: