Converting to Polar Coordinates:
In Exercises 29–-32, use polar coordinates to set up and evaluate the double
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Calculus: Early Transcendental Functions
- Use polar coordinates to set up and evaluate the double integral ∫R∫ f(x, y) dA. f(x, y) = arctan( y /x) R: x2 + y2 ≥ 1, x2 + y2 ≤ 4, 0 ≤ y ≤ xarrow_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_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. F = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forward
- Using Stokes' theorem, solve the line integral of G(x, y, z) - (1, x + yz, xy-√z) around the boundary of surface S, which is given by the piece of the plane 3x + 2y + z = 1 where x, y, and z all ≥ 0.arrow_forwardConverting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/(x2 + y2) over the region 1<= x2 + y2<= e^2.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
- 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_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_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forward
- How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface F(x, y, z) = 0? Of the surface which is the graph of z = ƒ(x, y)? Give examples.arrow_forwardDouble integral to line integral Use the flux form of Green’sTheorem to evaluate ∫∫R (2xy + 4y3) dA, where R is the trianglewith vertices (0, 0), (1, 0), and (0, 1).arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. F = sin 3y i + cos 7x j; C is the rectangle with vertices at (0, 0),(pi/7,0),(pi/7,pi/3) and (0,pi/3) a) 0 b) 2/3π c) - 2/3π d) -4/3 πarrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning