Math Problem Statement
Find the area of the surface. the part of the surface z = xy that lies within the cylinder x2 + y2 = 64
Solution
To find the area of the surface that lies within the cylinder , we can use surface area formulas from multivariable calculus.
Steps:
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Surface Area Formula: The general formula for the surface area of a parametric surface is: where is the projection of the surface onto the -plane.
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Partial Derivatives: Since , we first compute the partial derivatives:
The square of these derivatives are:
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Surface Area Integral: The surface area formula becomes: where is the disk defined by .
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Convert to Polar Coordinates: To simplify the integral, we convert to polar coordinates where and . The area element in polar coordinates is , and the equation .
So the integral becomes:
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Evaluate the Radial Integral: First, compute the radial part of the integral:
Let , so . This transforms the integral into:
Now calculate :
So the radial integral evaluates to:
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Complete the Surface Area Integral: Multiply by the angular part, which is :
Final Answer:
The surface area of the part of the surface that lies within the cylinder is approximately:
Would you like more details or have any questions?
Here are 5 related questions to consider:
- How do polar coordinates simplify surface area integrals?
- What is the geometric interpretation of the surface ?
- How do we compute surface area for surfaces in different coordinate systems?
- Why do we use the cylinder as the boundary in this problem?
- How do changes in surface height affect the total surface area?
Tip: Converting to polar coordinates is a powerful technique for symmetry-based problems, especially when the boundary is circular.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Surface Area
Polar Coordinates
Formulas
Surface area formula: A = ∬_D √(1 + (∂z/∂x)^2 + (∂z/∂y)^2) dA
Partial derivatives of z = xy: ∂z/∂x = y, ∂z/∂y = x
Surface area integral in polar coordinates: A = ∫_0^2π ∫_0^8 √(1 + r^2) r dr dθ
Theorems
Surface area of parametric surfaces
Conversion to polar coordinates for integrals
Suitable Grade Level
University level (Calculus III)
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