Find the area of the surface. The part of the paraboloid z = 1 - x2 - y2 that lies above the plane z = -6

The area of a shape is the amount of space it covers.

The surface area is approximately 81.3 unit squares

The given parameters are:

Calculate the partial derivatives of

So, the surface area is:

So, we have:

Factor out 4

Substitute

So, we have:

Multiply by 1

Express 1 as 8/8

Rewrite as:

From the question,

So, we have:

Add 6 to both sides

So, we have:

Integrate

Substitute

Rewrite as:

Integrate

So, we have:

Rewrite as:

Hence, the surface area is approximately 81.3 unit squares

Read more about surface areas at:

link

π/6 [(29)^³/₂ − 1]

81.247

Step-by-step explanation:

Surface area is:

S = ∫∫ √(fₓ² + fᵧ² + 1) dA

The partial derivatives are:

fₓ = -2x

fᵧ = -2y

Substituting:

S = ∫∫ √(4x² + 4y² + 1) dA

To make this easier, we can convert to polar coordinates.

S = ∫∫ √(4r² + 1) dA

S = ∫∫ √(4r² + 1) r dr dθ

S = ⅛ ∫∫ 8r √(4r² + 1) dr dθ

The limits for θ are 0 to 2π.  The minimum of r is 0.  The maximum of r is:

-6 = 1 − x² − y²

-6 = 1 − r²

r² = 7

r = √7

Integrating the first integral:

S = ⅛ ∫ ⅔ (4r² + 1)^³/₂ |₀ᴿ dθ

S = ⅛ ∫ [⅔ (4(7) + 1)^³/₂ − ⅔ (4(0) + 1)^³/₂] dθ

S = ⅛ ∫ [⅔ (29)^³/₂ − ⅔] dθ

S = ¹/₁₂ [(29)^³/₂ − 1] ∫ dθ

Integrating the second integral:

S = ¹/₁₂ [(29)^³/₂ − 1] θ |₀²ᵖⁱ

S = ¹/₁₂ [(29)^³/₂ − 1] (2π − 0)

S = π/6 [(29)^³/₂ − 1]

S ≈ 81.247

3: 2

step-by-step explanation:

shorter piece is 2: 3, so just reverse it, and you get 3: 2

arrange the equations with like terms in columns.

analyze the coefficients of x or y.

add the equations and solve for the remaining variable.

substitute the value into either equation and solve.

check the solution.

step-by-step explanation: