Integrate f(x,y,z)=z*sqrt(x^2+y^2) over the cylinder x^2+y^2=4 for (1 < or = to z < or = to 5)

Question

I am really stuck on this question. Any help will be greatly appreciated!

 

Integrate f(x,y,z)=z*sqrt(x^2+y^2) over the cylinder x^2+y^2=4 for (1 < or = to z < or = to 5)

Answer 1

 

Hi Billy,

 

∫∫∫⁵₁ z√(x²+y²) dzdydx

 

         = ∫∫ √(x²+y²)·(∫⁵₁ z dz) dydx

 

         = ∫∫ √(x²+y²)·((1/2)z²)|⁵₁ dydx

 

         = ∫∫ √(x²+y²)·((1/2)(5² - 1²)) dydx

 

         = 12∫∫ √(x²+y²) dydx                   (Eq. 1)

 

√(x²+y²) is the distance between (x, y) and the origin in the x-y plane, and

Eq. 1 can be calculated more easily in a polar coordinate system.

 

That is,

 

12∫∫√(x²+y²) dydx

 

         = 12∫^2π₀∫²₀ r · rdrdθ                      (Eq. 2)

 

where √(x²+y²) is replaced by r and

the double integration in a x-y plane is converted to that in a polar coordinate system

 

12∫^2π₀∫²₀ r · rdrdθ

 

         = 12∫^2π₀∫²₀ r² drdθ

 

         = 12∫^2π₀ (r³/3)|²₀ dθ

 

         = 4∫^2π₀ r³|²₀ dθ

 

         = 4∫^2π₀ (2³ - 0³) dθ

 

         = 32∫^2π₀ 1 dθ

 

         = 32· θ|^2π₀

 

         = 32· (2π - 0)

 

         = 64π