Concept explainers
Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the xy-plane and under the surface
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Calculus
- Deteremine the area between the curves x= y^2+1, x=5, y=-3, y=3.arrow_forwardStokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.arrow_forwardSurface integrals using a parametric description Evaluate the surface integral ∫∫S ƒ dS using a parametric description of the surface. ƒ(x, y, z) = y, where S is the cylinder x2 + y2 = 9, 0 ≤ z ≤ 3arrow_forward
- Surface integrals using a parametric description Evaluate the surface integral ∫∫S ƒ dS using a parametric description of the surface. ƒ(x, y, z) = x, where S is the cylinder x2 + z2 = 1, 0 ≤ y ≤ 3arrow_forwardThe volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone. The volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone? What is the x coordinate of the centroid of the volume?arrow_forwardSurface integrals using a parametric description Evaluate the surface integral ∫∫S ƒ dS using a parametric description of the surface. ƒ(x, y, z) = x2 + y2, where S is the hemisphere x2 + y2 + z2 = 36, for z ≥ 0arrow_forward
- Using Stokes’ Theorem to evaluate a line integral Evaluate the lineintegral ∮C F ⋅ dr, where F = z i - z j + (x2 - y2) k and C consists of the three line segments that bound the plane z = 8 - 4x - 2y in the first octant, oriented as shown.arrow_forwardStokes’ 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.arrow_forwardArea of the plane region y=2x^2+1 and y=x^2+5arrow_forward
- Using Stokes’ Theorem to evaluate a surface integral Evaluate∫∫S (∇ x F) # n dS, where F = -y i + x j + z k, in the following cases.a. S is the part of the paraboloid z = 4 - x2 - 3y2 that lies within the paraboloid z = 3x2 + y2 (the blue surface as shown). Assume n pointsin the upward direction on S.b. S is the part of the paraboloid z = 3x2 + y2 that lies within the paraboloidz = 4 - x2 - 3y2, with n pointing in the upward direction on S.c. S is the surface in part (b), but n pointing in the downward direction on S.arrow_forwardArea of Plane Region 2. R: y = 6x − x2and y = x2 − 2x.3. R: x2 + 3y = 4 and x − 2y = 4.4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forwardArea of Plane Region 4. R: x + 2y = 2, y− x = 1 and 2x + y = 7arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning