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.
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Chapter 15 Solutions
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- 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_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_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) = x2 + y2, where S is the hemisphere x2 + y2 + z2 = 36, for z ≥ 0arrow_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_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
- Finding the Volume of a Solid In Exercises 17-20, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4.y =1/2x3, y = 4, x = 0arrow_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_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_forward
- Surface areas Use a surface integral to find the area of the following surfaces. The hemisphere x2 + y2 + z2 = 9, for z ≥ 0arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluatethe line integral in Stokes’ Theorem to determine the value of thesurface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upwarddirection. F = ⟨x, y, z⟩; S is the upper half of the ellipsoid x2/4 + y2/9 + z2 = 1.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_forward
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