Finding the Volume Under a Paraboloid Using Polar Coordinates
To find the volume of the solid that lies beneath the paraboloid given by the equation z = 8 – 2x² – 2y² and above the xy-plane, we can simplify our calculations by switching to polar coordinates. This method helps to handle circular symmetry efficiently.
Step 1: Understanding the Surface and Volume
The paraboloid opens downward and intersects the xy-plane when z = 0:
0 = 8 – 2x² – 2y²
Simplifying this, we find:
2x² + 2y² = 8
x² + y² = 4
This equation describes a circle of radius 2 on the xy-plane. Thus, our volume calculation will be constrained within this circle.
Step 2: Converting to Polar Coordinates
In polar coordinates, we substitute:
x = r cos(θ)
y = r sin(θ)
Then, the equation for z becomes:
z = 8 – 2(r cos(θ))² – 2(r sin(θ))²
z = 8 – 2r²(cos²(θ) + sin²(θ))
Since extit{cos²(θ) + sin²(θ) = 1}, we simplify to:
z = 8 – 2r²
Step 3: Setting Up the Volume Integral
The volume V can be calculated using the double integral in polar coordinates:
V = ∫∫_D z \, r \, dr \, dθ
Where D is the circular area defined by the radius 2:
0 ≤ r ≤ 2
0 ≤ θ ≤ 2π
Substituting for z gives:
V = ∫₀²π ∫₀² (8 – 2r²) r \, dr \, dθ
Step 4: Evaluating the Integral
Now we compute the inner integral:
∫₀² (8r – 2r³) \, dr
Calculating the inner integral:
= [4r² – 0.5r⁴]₀²
= [4(2)² – 0.5(2)⁴] – 0
= [16 – 8] = 8
Now, substituting back into the outer integral:
V = ∫₀²π 8 \, dθ
This evaluates to:
= 8(2π) = 16π
Conclusion
Hence, the volume of the solid that lies underneath the paraboloid and above the xy-plane is:
V = 16π cubic units.