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Finding Surface AreaIn Exercises 3–16, find the area of the surface given by
R: square with vertices
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Chapter 14 Solutions
Calculus
- A). Use Pappus's theorem for surface area and the fact that the surface area of a sphere of radius d is 4pid^2 to find the centroid of the semicircle x=(d^2-y^2)^0.5arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The wedge above the xy-plane formed when the cylinder x2 + y2 = 4 is cutby the planes z = 0 and y = -z.arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The wedge bounded by the parabolic cylinder y = x2and the planes z = 3 - y and z = 0.arrow_forward
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by x = 0, x = 2, y = z, y = z + 1, z = 0, and z = 4arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The solid bounded by the surfaces z = ey and z = 1 over the rectangle{(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ ln 2}arrow_forwardSurface areas Use a surface integral to find the area of the following surfaces. The surface ƒ(x, y) = √2 xy above the polar region{(r, θ): 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π}arrow_forward
- Variable-density solids Find the coordinates of the center of mass of the following solid with variable density. R = {(x, y, z): 0 ≤ x ≤ 4, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1};ρ(x, y, z) = 1 + x/2arrow_forwardVolumes of solids Use a triple integral to find the volume of thefollowing solid. The wedge in the first octant bounded by the cylinder x = z2 andthe planes z = 2 - x, y = 2, y = 0, and z = 0arrow_forwardDeteremine the area between the curves x= y^2+1, x=5, y=-3, y=3.arrow_forward
- Stokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forwardSurface integrals using an explicit description Evaluate the surface integral ∫∫S ƒ(x, y, z) dS using an explicit representation of the surface. ƒ(x, y, z) = x2 + y2; S is the paraboloid z = x2 + y2, for 0 ≤ z ≤ 1.arrow_forwardSurface areas Use a surface integral to find the area of the following surfaces. The hemisphere x2 + y2 + z2 = 9, for z ≥ 0arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning