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Concept explainers
Evaluating line
a. Use a parametric description of C to evaluate the integral directly.
b. Use the Fundamental Theorem for line integrals.
31. φ(x, y, z) = (x2 + y2 + z2)/2; C: r(t) = 〈cos t, sin t, t/π〉, for 0 ≤ t ≤ 2π
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Chapter 17 Solutions
Calculus: Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)
Additional Math Textbook Solutions
Glencoe Math Accelerated, Student Edition
University Calculus: Early Transcendentals (4th Edition)
University Calculus: Early Transcendentals (3rd Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
- Use Green's Theorem to evaluate f, F •dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y cos x xy sin x, xy + x cos x), C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)arrow_forward(a) Let f(2) = 24 + 5z3 Evaluate the integral Of(2)dz, where the contour C is the circle z = 2.arrow_forward²/F F. dr. (Check the orientation of the curve before applying the theorem.) F(x, y) = (y cos(x) - xy sin(x), xy + x cos(x)), C is the triangle from (0, 0) to (0, 6) to (3, 0) to (0, 0) Use Green's theorem to evaluatearrow_forward
- Stokes' Theorem (1.50) Given F = x²yi – yj. Find (a) V x F (b) Ss F- da over a rectangle bounded by the lines x = 0, x = b, y = 0, and y = c. (c) fc ▼ x F. dr around the rectangle of part (b).arrow_forwardUse Green's Theorem to evaluate fa F. dr where F(x, y) = (y cos x, x² + 2y sin x) and C is the triangle from (0,0) to (2, 6), (2,6) to (2,0), and from (2, 0) back to (0,0). Make sure to check the orientation of the curve before applying Green's Theorem.arrow_forward2.Proof that any tangent plane for the surface F( F) point = 0 passses through a fixedarrow_forward
- Sketch the curve given in parametric form by r = ti+ sin tj + cos tk, 0 < t < 2π, and write down an integral to determine its length. Calculate the length of this curve. Write down a simple algebraic relation that determines a simple surface on which the curve lies. The length of this curve is: (a) 2π√2 (b) 2π (c) 2√2 (d) π/2√√2 The surface on which the curve lies is: (i) a sphere x² + y²+z² = 1 (ii) a cylinder x² + y² = 1 (iii) an elliptic cylinder (3x)2 + y² = 1 (iv) a cylinder y²+z² = 1 22arrow_forwardQ 2. Let C₁ be the straight line from the point (1,0) to the point (0, 1) in Figure 1. Let C₂ be an oriented and closed path in Figure 1. (a) (b) Evaluate the line integral of F = 4xi + 2xj along C₁. Evaluate the line integral of F = sin(2x)i + ej along C₂. Figure 1: A closed and oriented patharrow_forwardFind an equation for the tangent plane to the surface z + 4 = xy cos(z) at the point (4, 1, 0). Submit Questionarrow_forward
- Use Green's Theorem to evaluate F. dr, where F(x, y) = (y - cos y, x sin y) and C is the circle (x − 3)² + (y + 4)² = 4 oriented clockwise. (Check the orientation of the curve before applying the theorem.)arrow_forwardUse Green's Theorem to evaluate So F. dr, where F =(√x+4y, 3x + √√ÿ) C consists of the arc of the curve y = 1x - x² from (0,0) to (1,0) and the line segment from (1,0) to (0,0).arrow_forwardF. dr. (Check the orientation of the curve before applying the theorem.) Jc Use Green's Theorem to evaluate F(x, y) = (y2 cos(x), x² + 2y sin(x)) C is the triangle from (0, 0) to (1, 3) to (1, 0) to (0, 0)arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning
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