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39. Find the surface area of the portion of the hyperbolic paraboloid r(u, v) = (u+ v) i + (u – v) j+ uvk for which u? + u < 4.

39. Find the surface area of the portion of the hyperbolic paraboloid r(u, v) = (u+ v) i + (u – v) j+ uvk for which u? + u < 4.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem 39: Surface Area of a Hyperbolic Paraboloid** Find the surface area of the portion of the hyperbolic paraboloid given by the parameterization: \[ \mathbf{r}(u, v) = (u + v)\mathbf{i} + (u - v)\mathbf{j} + uv\mathbf{k} \] for which \( u^2 + v^2 \leq 4 \). **Explanation:** This problem involves calculating the surface area of a hyperbolic paraboloid, a type of doubly ruled surface. The parameterization given allows us to explore the surface defined by vector functions in terms of the parameters \( u \) and \( v \). The constraint \( u^2 + v^2 \leq 4 \) describes a circular domain of radius 2 in the \( uv \)-plane, which determines the portion of the hyperbolic paraboloid to be analyzed. The task requires applying methods of multivariable calculus, such as surface integrals, to find the surface area over this domain. This parameterization brings out the intricate interplay between the parameters and the 3D surface geometry.
Transcribed Image Text:**Problem 39: Surface Area of a Hyperbolic Paraboloid** Find the surface area of the portion of the hyperbolic paraboloid given by the parameterization: \[ \mathbf{r}(u, v) = (u + v)\mathbf{i} + (u - v)\mathbf{j} + uv\mathbf{k} \] for which \( u^2 + v^2 \leq 4 \). **Explanation:** This problem involves calculating the surface area of a hyperbolic paraboloid, a type of doubly ruled surface. The parameterization given allows us to explore the surface defined by vector functions in terms of the parameters \( u \) and \( v \). The constraint \( u^2 + v^2 \leq 4 \) describes a circular domain of radius 2 in the \( uv \)-plane, which determines the portion of the hyperbolic paraboloid to be analyzed. The task requires applying methods of multivariable calculus, such as surface integrals, to find the surface area over this domain. This parameterization brings out the intricate interplay between the parameters and the 3D surface geometry.
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