Calculus: Vector Integration

Question

Three paths from (0,0) to (1,2) are defined by

(a) y=2x

(b) y=2x^2

(c) y=0 from (0,0) to (1,0) and x= 1 from (1,0) to (1,2).

Sketch each path and along each path find (integral)c Fdr, where F= y^2i+xyj

Answer 1

Fdr = (y²i + xyj)(dxi + dyj) = y²dx + xydy;

∫_lFdr = ∫_l(y²dx + xydy) where l is contur of integration.

a) l is line y = 2x, then dy = 2dx and 0≤x≤1; Then our integrl ∫_lFdr = ∫₀¹(4x²dx +4x²dx) = ∫₀¹8x²dx =8/3x³₀¹ = 8/3.

b) l is line y = 2x², then dy = 4xdx and 0≤x≤1. Then our integral ∫_lFdr = ∫₀¹(4x⁴ + 8x⁴)dx = (12x⁵/5)₀¹ = 12/5;

c) l is l₁ y = 0 ,dy = 0, 0≤x≤1 and l₂ x = 1, dx = 0 and 0≤y≤2.

Then ∫_lFdr = ∫₀²ydy = y²/2₀² = 2