The Triple Integral

1. The Triple Integral

2. Remark The triple integral of f over the region { (x,y,z) : x є [ a , b ] , α(x) ≤ y ≤ β(x) , g(x,y) ≤ z ≤ h(x,y) } is equal to: abα(x)β(x)g(x,y)h(x,y)f(x,y,z)dz dy dx = ab[α(x)β(x)[ g(x,y)h(x,y)f(x,y,z) dz ] dy ] dx

3. Example 1 • Evaluat the triple integral I of the given function f over the given region D. f(x,y,z) = 8xyz D = { (x,y,z) : x є [ 0, 3 ] , y є [ 0 , 2] , zє [ 0, 1 }

4. Solution I = 0302018xyzdz dy dx = 0302[ ( 4xyz2)│z = 1 - ( 4xyz2)│z = 0 ] dy dx = 03024xy dy dx = 03[ ( 2xy2)│y= 2 - ( 2x y2)│y= 0 ] dx = 038x dx = ( 4 x2)│x= 3 - ( 4 x2)│x= 0 = 36

5. Example 1 • Evaluat the triple integral I of the given function f over the given region D. f(x,y,z) = z D = { (x,y,z) : x є [ 0, 2 ] , 0 ≤ y ≤ 3x , 0 ≤ z ≤ xy }

6. Solution I = 0203x0xyzdz dy dx = 0203x[ (z2 / 2)│z = xy - (z2 / 2 )│z = 0 ] dy dx = (1/2) 0203xx2 y2 dy dx = (1/6) 02[ (x2y3)│y= 3x - (x2y3)│y= 0 ] dx = (1/6) 0227 x5 dy dx = ?

7. Triple Integral in Cylindrical Coordinates Let g( r , θ ) ≤ h( r ,θ) , K(θ) ≤ M(θ) ; α ≤ θ ≤ β Let D be the region ( in the cylindrical coordinates) defined by: D = { (r,θ ,z) : α ≤θ ≤ β , K(θ) ≤ r ≤ M(θ) , g(r,θ) ≤ z ≤ h(r,θ) } If f is continuous in x = cosθ , y = rsinθ and z Then the triple integral of f over D is : ∭D f( rcosθ , rsinθ , z ) r dz dr dθ = αβK(θ)M(θ)g(r,θ)h(r,θ)f( rcosθ , rsinθ, z ) rdz dr dθ

8. Example • Let f(x,y,z) = z and D = { (r,θ,z) : 0 ≤ θ ≤ 2π , 0 ≤ r ≤ 2 , 0 ≤ z ≤ √ 16 - r2 } Evaluate the triple integral of f over D

9. Solution I = 02π020√16 – r2z rdz dr dθ = (1/2)02π02 [ (z2 r )│z =√16 – r2 - (z2 r )│z = 0 ] dr dθ = (1/2)02π02( 16 -r2) r dr dθ = (1/2)02π02( 16r -r3) r dr dθ = (1/2)02π[( 8 r2 - (1/4)r4)│r = 2 - (8 r2 - (1/4)r4)│r = 0 ] dθ = 02π14 dθ = 28π

10. Triple Integral in Spherical Coordinates Let D be the region ( in the spherical coordinates) defined by: D = { ( ρ , θ , φ ) : α ≤θ ≤ β , γ ≤ φ ≤ ω , a ≤ ρ≤ b } If f is continuous in: x = ρ cosθ sinφ , y = ρ sinθ sinφ and z = ρ cosφ Then the triple integral of f over D is : ∭D f( ρ cosθ sinφ , ρ sinθ sinφ , ρ cosφ ) ρ2 sinφ dρ dφ dθ = αβγωabf( ρ cosθ sinφ , ρ sinθ sinφ , ρ cosφ) ρ2 sinφ dρ dφ dθ

11. Example • Let f be a constant function with the value 5 on R3 Find the triple integral of f on the following region D = { (ρ,θ,φ) : α ≤θ ≤ π, 0≤ φ ≤π/3, 0≤ ρ≤3 }

12. Solution I= 0π0π/3035 ρ2 sinφ dρ dφ dθ = (5/3) 0π0π/3[ ( ρ3sinφ) │ρ = 3 - ( ρ3sinφ) │ρ = 0 ]dφ dθ = (5/3) 0π0π/327 sinφ dφ dθ = 45 0π0π/3[ ( - cosφ )│ φ = π/3 - ( - cosφ )│ φ = 0 ] dθ = - 45 0π[ (1/2) - 1 ] dθ = - 45 0π( - ½ ) dθ = (45/2 ) [( θ )│ θ = π - ( θ )│ θ = 0 = 45π/2

13. Example 2 Let f(x,y,z) = (x2 + y2 + z2 ) ½ And let D be the region in the first octant in R3 and bounded by the graphs: 4 – z = 9x2 + y2 , y = 4x , z = 0 and y = 0 . Graph the region D and set up the triple integral of the function f over D.

14. Solution • x2 = 1/9 [ 4 – y2 ] → x = (⅓) √4 – y2 • z = 4 – 9x2 – y2 • I = ∫∫∫D f(x,y,z) dz dx dy = 0∫8/5y/4∫ (1/3(√4 – y2)0∫ 4 - 9 x2 – y2 (x2 + y2 +z2 )1/2 dz dy dx