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Line integrals

insight into calculus <br>

Nosmilo
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Line integrals

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  1. Line integrals May 6, 2025 Line integrals May 6, 2025 1/14

  2. Line integrals Line integrals May 6, 2025 2/14

  3. Line integrals May 6, 2025 3/14

  4. Theorem: (Evaluation of line integral in R2) Let C be a smooth curve defined by the parametric equations: a ≤ t ≤ b. x = x(t), y = y(t), If f (x,) is continuous on a region D in R2containing C, then Z Z Z Zb Zb Zb p (x′(t))2+ (y′(t))2dt f (x,y)ds = f (x(t),y(t)) C a f (x(t),y(t))(x′(t))dt f (x,y)dx = C a f (x(t),y(t))(y′(t))dt f (x,y)dy = C a Line integrals May 6, 2025 4/14

  5. Theorem: (Line integral over piecewise smooth curves) Suppose C is a piecewise smooth curve, that is C is a union of finite number of smooth curves, C1,C2,...,Cn, joined end-to-end (that is the initial point of Ci+1 is the terminal point of Ci), and f is continuous in D ⊂ R2containing C, then the integral of f along C is defined as Z Z Z Z f (x,y)ds = f (x,y)ds + f (x,y)ds + ... + f (x,y)ds. C C1 C2 Cn Line integrals May 6, 2025 5/14

  6. Some common curves and their parameterization Table: Some common curves and their parameterization Curves Circle: x2+ y2= a2 Ellipse:x2 Parameterization x(t) = acos(t), y(t) = b sin(t), t ∈ [0,2π] x(t) = acos(t), y(t) = b sin(t), t ∈ [0,2π] x(t) = t, y(t) = f (t) x(t) = g(t), y(t) = t x(t) = x0+ t(x1− x0), y(t) = y0+ t(y1− y0), t ∈ [0,1] x(t) = x0+ t(x1− x0),y(t) = y0+ t(y1− y0) z(t) = z0+ t(z1− z0), t ∈ [0,1] a2+y2 b2= 1 y = f (x) x = g(y) Line segment from (x0,y0) to (x1,y1) Line segment from (x0,y0,z0) to (x1,y1,z1) Line integrals May 6, 2025 6/14

  7. Z (2 + x2y)ds, where C is the upper half of the unit Example: Evaluate C circle x2+ y2= 1. Line integrals May 6, 2025 7/14

  8. Z 2xds, where C is the parabola y = x2from (0,0) to Example: Evaluate C (1,1). Line integrals May 6, 2025 8/14

  9. Z (x + 2y)dx + (x − 2y)dy, where C is the line segment from (1,1) to (3,−1). Example: Evaluate C Line integrals May 6, 2025 9/14

  10. Z 6x2ydx + xydy, where C is the graph y = x3+ 1 Example: Evaluate C from (−1,0) to (1,2). Line integrals May 6, 2025 10/14

  11. Z Example: Evaluate 2xds, where C consists of the arc C1of the parabola C y = x2from (0,0) to (1,1) and the line segment C2from (1,1) to (1,2). Line integrals May 6, 2025 11/14

  12. Line integrals May 6, 2025 12/14

  13. Line integrals May 6, 2025 13/14

  14. Line integrals May 6, 2025 14/14

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