This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To evaluate the work done by a vector field F along a curve C, we compute the line integral _C F · dr. Step 1: Parameterize the line segment C. The line segment goes from P_0(2,0) to P_1(0,2). We can parameterize it as r(t) = (1-t)P_0 + tP_1 for 0 t 1. r(t) = (1-t) 2,0 + t 0,2 r(t) = 2(1-t), 0 + 0, 2t r(t) = 2-2t, 2t So, x(t) = 2-2t and y(t) = 2t. Step 2: Calculate dr. First, find the derivative of r(t): r'(t) = (d)/(dt)(2-2t), (d)/(dt)(2t) = -2, 2 Then, dr = r'(t) \, dt = -2, 2 \, dt. Step 3: Express the vector field F(x,y) in terms of t. The given vector field is F(x,y) = y-x, x . Substitute x=2-2t and y=2t: F(x(t), y(t)) = (2t) - (2-2t), (2-2t) F(x(t), y(t)) = 2t - 2 + 2t, 2-2t F(x(t), y(t)) = 4t - 2, 2-2t Step 4: Compute the dot product F · dr. F · dr = 4t - 2, 2-2t · -2, 2 \, dt F · dr = ((4t - 2)(-2) + (2-2t)(2)) \, dt F · dr = (-8t + 4 + 4 - 4t) \, dt F · dr = (-12t + 8) \, dt Step 5: Evaluate the definite integral. The work done is _C F · dr = _0^1 (-12t + 8) \, dt. _0^1 (-12t + 8) \, dt = [ -12(t^2)/(2) + 8t ]_0^1 = [ -6t^2 + 8t ]_0^1 Now, evaluate at the limits of integration: = (-6(1)^2 + 8(1)) - (-6(0)^2 + 8(0)) = (-6 + 8) - (0) = 2 The work done is 2. Drop the next question.