Vector Calc 8-18-2020 (Part 2)

Calculus Level pending

Consider a straight line segment from ( x 1 , y 1 , z 1 ) = ( 0 , 0 , 0 ) (x_1, y_1, z_1) = (0,0,0) to ( x 2 , y 2 , z 2 ) = ( 5 , 4 , 3 ) (x_2, y_2, z_2) = (5,4,3) . Determine the value of the line integral of the vector field F ⃗ = ( F x , F y , F z ) = ( 5 y , 4 x 2 , 3 z ) \vec{F} = (F_x, F_y, F_z) = (5 y , 4 x^2, 3 z) over the line segment.

∫ F ⃗ ⋅ d ℓ ⃗ \int \vec{F} \cdot \vec{d \ell}


The answer is 196.83.

Steven Chase by Steven Chase , 37, USA

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1 solution

Ron Gallagher
Aug 18, 2020

The line segment is given by the parametric equations x = 5t, y = 4t, z = 3t for t in [0,1]. We have F(l(t)) = (20t, 100*t^2, 9t). Hence:

F(l(t)).(dl/dt) = 127 t + (400/3) t^2.

Integrating this function over [0,1] gives 127/2 + (400/3), which is approximately 196.83

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