Evaluate the line integral where C is the given curve. ?C x^2dx + y^2dy + z^2dz

Question

Evaluate the line integral where C is the given curve. ?C x^2dx + y^2dy + z^2dz C consists of the line segments from (0,0,0) to (1,1,-2) and from (1,1,-2) to (4,1,0)

Explanation / Answer

Parameterize each line segment: (i) (0, 0, 0) to (1, 2, -1) ==> c(t) = t * (1,2,-1) for t in [0,1]. ==> (x,y,z) = (t, 2t, -t). So, (dx,dy,dz) = (1 dt, 2 dt, -1 dt). Thus, ? x^2 dx + y^2 dy + z^2 dz = ?(t = 0 to 1) t^2 (1 dt) + (2t)^2 (2 dt) + (-t)^2 (-dt) = ?(t = 0 to 1) 8t^2 dt = 8/3. (ii) (1, 2, -1) to (2, 2, 0) ==> c(t) = (1,2,-1) + t * (2-1, 2-2, 0 -(-1)) for t in [0,1]. ==> (x,y,z) = (1 + t, 2, -1 + t). So, (dx,dy,dz) = (1 dt, 0 dt, 1 dt). Thus, ? x^2 dx + y^2 dy + z^2 dz = ?(t = 0 to 1) (1 + t)^2 (1 dt) + (2)^2 (0 dt) + (-1 + t)^2 (1 dt) = ?(t = 0 to 1) 2t^2 + 2 dt = 8/3. Adding the contributions from (i) and (ii): ?C x^2 dx + y^2 dy + z^2 dz = 8/3 + 8/3 = 16/3.

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