Suppose that surface σ is parameterized by r(u, v)=⟨ucos(3v),usin(3v),v⟩, 0≤u≤7 and 0≤v≤2π3 and f(x, y,z)=x2+y2+z2. set up the surface integral (you don't need to evaluate it). ∫σ∫f(x, y,z)ds=∫r∫f(x(u, v),y(u, v),z(u, v))∥∥∥∂r∂u×∂r∂v∥∥∥da = 2pi/3 ∫ 0 7 ∫ 0 u^2+v^2 ∥∥∥ ∥∥∥dudv

Looks like you have most of the details already, but you're missing one crucial piece.

is parameterized by

for and , and a normal vector to this surface is

with norm

So the integral of is

13

step-by-step explanation:

169-1=168

168-3=165

165-5=160

160-7=153

153-9=144

144-11=133

133-13=120

120-15=105

105-17=88

88-19=69

69-21=48

48-23=25

25-25=0

since it took 13times to completely subtract it hence its square root is 13