Concept explainers
EXPLORING CONCEPTS
Using Different Methods Evaluate
where S is the first-octant portion of the plane
by projecting 5 onto (a) the xy-plane, (b) the xz-plane, and (c) the yz-plane. Verify that all answers are the same.
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Chapter 15 Solutions
Calculus (MindTap Course List)
- Surface area using an explicit description Find the area ofthe following surfaces using an explicit description of the surface. The part of the hyperbolic paraboloid z = 3 + x2 - y2 above thesector R = {(r, θ): 0 ≤ r ≤ √2, 0 ≤ θ ≤ π/2}arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The plane 2x - 4y + 3z = 16arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The portion of the cylinder x2 + y2 = 9 in the first octant, for 0 ≤ z ≤ 3arrow_forward
- Describe and sketch the surface in double-struck R3 represented by the equation y = 2x. The equation represents the set of all points in double-struck R3 whose y-coordinate is ( insert answer) times their x-coordinate, that is, {(x, 2x, z) | x is in double-struck R, z is in double-struck R}. This is a vertical plane that intersects the xy-plane in the line y = , z = .The portion of this plane that lies in the first octant is sketched in the figure.arrow_forwardSurface area using a parametric description Find the area ofthe following surfaces using a parametric description of the surface. The plane z = 10 - x - y above the square | x | ≤ 2, | y | ≤ 2arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cap of the sphere x2 + y2 + z2 = 16, for 2√2 ≤ z ≤ 4arrow_forward
- Parametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cylinder y2 + z2 = 36, for 0 ≤ x ≤ 9arrow_forwardQuestion: By using normals, analyze the geometric relationship of the systems of equations. Determine the intersection of the planes, if it exists. If the intersection is a line, express your answer in parametric form. a) P₁ : 2x + y + z = 6 P₂ : x - y - z = -9 P₃ : 3x + y = 2 [One POI (-1,5,3)] b) P₁ : 2x - y + 2z = 2 P₂ : 3x + y - z = 1 P₃ : x - 3y + 5z = 4 [Show Co-planar-Triangular Prism] c) P₁ : 2x + y - z = 0 P₂ : x - 2y + 3z = 0 P₃ : 9x + 2y = 0 [Intersects at Line-State Line]arrow_forwardDetermine the location of a point (x, y, z) that satisfies the condition. x > 0 a) The point is behind the yz-plane. b) The point is behind the xz-plane. c) The point is in front of the yz-plane. d)The point above the xy-plane. e)The point below the xy-plane. f)The point is in front of the xz-plane.arrow_forward
- Parametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The frustum of the cone z2 = x2 + y2, for 2 ≤ z ≤ 8arrow_forwardIdentifying surfaces Identify the following surfaces by name. y2 - z2 = 2arrow_forwardParametric descriptions Give a parametric description of the form r(u, v) = ⟨x(u, v), y(u, v), z(u, v)⟩ for the following surfaces.The descriptions are not unique. Specify the required rectangle in the uv-plane. The cone z2 = 4(x2 + y2), for 0 ≤ z ≤ 4arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage