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Finding Surface AreaIn Exercises 3–16, find the area of the surface given by
R: triangle with vertices
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Calculus (MindTap Course List)
- 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_forwardA). 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 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 wedge above the xy-plane formed when the cylinder x2 + y2 = 4 is cutby the planes z = 0 and y = -z.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_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_forward
- Volumes of solids Use a triple integral to find the volume of thefollowing solid. The solid between the sphere x2 + y2 + z2 = 19 and the hyperboloidz2 - x2 - y2 = 1, for z > 0arrow_forwardSurface areas Use a surface integral to find the area of the following surfaces. The hemisphere x2 + y2 + z2 = 9, for z ≥ 0arrow_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_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 solid bounded by x = 0, y = z2, z = 0, and z = 2 - x - yarrow_forwardSetup, but don't evaluate, the integrals which give the volume of the solid formed by revolving the region bounded by y = x2+1, y = x, x = 1, x = 2 about these lines: a) x-axis b) y = -1 c) y = 6 d) y-axis e) x = -3 f) x = 4 g) x = 1arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning