Finding the Surface Area of the Sphere Above the Cone
The problem requires calculating the surface area of a region on the sphere where it is above the cone. Let’s break down the steps needed to solve this in detail.
1. Understand the Geometry
We have a sphere defined by the equation:
x² + y² + z² = 81
This means the sphere has a radius of:
R = √81 = 9
The cone is defined by:
z = √(x² + y²)
This is a right circular cone opening upwards along the z-axis.
2. Set the Limits
To find the surface area of the sphere above the cone, we need to determine where the sphere and the cone intersect. This occurs when:
z = √(81 – x² – y²)
Setting the equations equal gives us:
√(x² + y²) = √(81 - x² - y²)
Squaring both sides leads to:
x² + y² = 81 - x² - y²
Thus:
2(x² + y²) = 81
From this, we find:
x² + y² = 40.5
3. Change to Polar Coordinates
To simplify calculations, we can convert to polar coordinates:
x = r cos(θ), y = r sin(θ)
In polar coordinates, the equation of the surface area becomes:
r² = 40.5
We also express the height of the sphere above the cone:
z = √(81 - r²)
4. Calculate Surface Area Using the Surface Integral
The surface area element on a sphere can be calculated using the formula:
dS = (R²) sin(φ) dφ dθ
Where:
- R = radius of the sphere
- φ = azimuthal angle (from the vertical)
- θ = polar angle (around the z-axis)
5. Integrating the Area
In our case, we will integrate:
S = ∫∫ R² sin(φ) dφ dθ
for φ ranging from 0 (the pole) to the angle where the sphere intersects the cone. We determine this angle as:
tan(φ) = (r sin(θ))/(√(x² + y²)) = 1
Evaluating through the limits gives us:
φ = π/4
6. Final Integration Bounds and Calculation
The θ bounds are from 0 to 2π and φ bounds from 0 to π/4. Thus, we finalize our surface area integral:
S = ∫(θ=0 to 2π) ∫(φ=0 to π/4) R² sin(φ) dφ dθ
The calculation of this integral will yield the surface area of the sphere above the cone. Using the integration, you can obtain the final result which will be expressed in square units.
Conclusion
Finding the surface area of the area of the sphere above the cone requires understanding the geometry involved, converting to an appropriate coordinate system, and correctly evaluating the surface integral based on defined limits. This process ultimately provides the solution to our original question.