Find the total charge Q of the wire if the line density is distributed along the

Question

Find the total charge Q of the wire if the line density is distributed along the wire according to the formula: q(x,y,z) = x^2 + y^2 - z^2. The vector equation of the wire is r = 4cos(t)i + (3t - 5)j + 4sin(t)k, where 0 < t < 3.

Explanation / Answer

Find the total charge Q of the wire if the line density is distributed along the wire according to the formula: q(x,y,z) = x^2 + y^2 - z^2. The vector equation of the wire is r = 4cos(t) i + (3t - 5) j + 4sin(t)k, where 0 dx = - 4 sin(t) dt y= (3t - 5) ==> dy = 3 dt z= 4sin(t) ==> dz =4cos(t) dt dr = - 4sin(t) dt i + 3dt j + 4cos(t) dtk |dt| = sqrt(4+9) dt = sqrt(13) q(x,y,z) = x^2 + y^2 - z^2 ==> q(t) = ( 4cos(t) )^2 + ( 3t - 5)^2 - ( 4sin(t))^2 int_{t=0}^{3} q(t) |r'(t)| dt = sqrt(13) int_{t=0}^{3} ( ( 4cos(t) )^2 + ( 3t - 5)^2 - ( 4sin(t))^2) dt = sqrt(13) ( 21 + 8 Sin(6) ) ANSWER

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