Concept explainers
Finding Surface Area In Exercises 37-42, find the area of the surface over the given region. Use a computer algebra system to verify your results.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Multivariable Calculus
- How do you find the area of a region 0 ≤ r1(θ) ≤ r ≤ r2(θ),a≤ θ ≤ b, in the polar coordinate plane? Give examples.arrow_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_forwardUsing 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_forward
- Surface 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_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) = x, where S is the cylinder x2 + z2 = 1, 0 ≤ y ≤ 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) = x2 + y2, where S is the hemisphere x2 + y2 + z2 = 36, for z ≥ 0arrow_forwardEvaluating a Surface Integral. Evaluate ∫∫ f(x, y, z)dS, where S f(x,y,z)=√(x2+y2+z2), S:x2+y2 =9, 0⩽x⩽3, 0⩽y⩽3, 0⩽z⩽9.arrow_forwardRadial fields and zero circulation Consider the radial vectorfields F = r/ | r | p, where p is a real number and r = ⟨x, y, z⟩ .Let C be any circle in the xy-plane centered at the origin.a. Evaluate a line integral to show that the field has zero circulation on C.b. For what values of p does Stokes’ Theorem apply? For those values of p, use the surface integral in Stokes’ Theorem to show that the field has zero circulation on C.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.arrow_forwardSurface areas Use a surface integral to find the area of the following surfaces. The hemisphere x2 + y2 + z2 = 9, for z ≥ 0arrow_forwardUsing 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_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning