93ç-ç‚Bø ‚5

93ç-ç‚Bø ‚5

1. Find the volume of the solid enclosed by the ellipsoid x²+2xy+5y²+4yz+4z² = 1.

Solution:

(x + y)² + (2y + z)²+√ 3z²

= 1 .

u= x + y, v = 2y + z, w =√ 3z

. x = u − v

2+ w 2√ 3 y = v

2 − w 2√ 3 z = w

√3

∂(x, y, z)

∂(u, v, w) =           

1 −1 2

1 2√

3 0 1

2 − 1

2√ 3

0 0 1

√3           

= 1

2√ 3

V = Z Z Z

Exyz

dV = Z Z Z

Euvw

∂(x, y, z)

∂(u, v, w)    

dV = 1 2√

3 ·4

3π = 2π 3√ 3

2. Let

F(x, y, z) =



2x + 3y⁴+ 2xyze^x2 + 2x 1 + x² + z²

 i + 

12xy³+ ze^x2 j+



ye^x2 + 2z 1 + x²+ z²

 k (a) Find a function f (x, y, z) such that ∇f = F.

(b) Evaluate the line integral Z

C

F · dr along the curve C given by r(t) = cos t, sin t,₂^t_π, 0 ≤ t ≤ 2π.

3. Evaluate the line integral Z

C

−y dx + (x² + y²− x) dy

(x − 1)²+ y² along the curve C which is defined by the polar equation r = 1

2 + cos θ, 0 ≤ θ ≤ 2π, and is oriented by increasing θ angle. You must draw the curve C first.

4. Suppose that a thin spiral ramp (helicoid) H, given by the parametric equation r(u, v) = hv cos u, v sin u, ui, 0 ≤ u ≤ 2π, 0 ≤ v ≤ 2, has density function ρ(x, y, z) = zpx²+ y².

1

(a) Find the equation of the tangent plane to H at (x, y, z) =

√3 2 ,3

2,π 3

! . (b) Find the mass of H.

Solution:

(a)

∂r

∂u = (−v sin u, v cos u, 1)

∂r

∂v = (cos u, sin u, 0)

∂r

∂u× ∂r

∂v = (− sin u, cos u, −v)

√3 2 ,3

2,π 3

!

= rπ 3,√

3

 ∂r

∂u × ∂r

∂v

   (^π3,√

3)

= −

√3 2 ,1

2,−√ 3

!

Equation of tangent plane:

x−

√3 2

!

−

√3 2

! +

 y− 3

2



· 1 2+

z− π 3

· (−√ 3) = 0

⇐⇒ −

√3 2 x+ y

2 −√

3z + π

√3 = 0

(b) k ∂r

∂u × ∂r

∂v k=√ 1 + v² The mass equals

Z Z

S

ρ dσ= Z ²π

0

Z ²

0

uv√

1 + v²dv du

= Z ²π

0

u

"

1

3 1 + v^2
32

2

0

# du

= 1 3

Z ²π 0

5³² − 1

u du= 2π² 3

5³² − 1

5. Find the flux of F(x, y, z) = ln x² + y²
i − 2z

x tan⁻¹ y x



j+ zp

x² + y²kacross the boundary surface Γ of D = {(x, y, z) : 1 ≤ x²+ y² ≤ 2, −1 ≤ z ≤ 2}. Here Γ is oriented by the outward normal to D.

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