Vector Calc 8-18-2020

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 , 4 y , 3 z 2 ) \vec{F} = (F_x, F_y, F_z) = (5 , 4y, 3 z^2) over the line segment.

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


The answer is 84.0.

Steven Chase by Steven Chase , 37, USA

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1 solution

Let the parametric equation of the line segment joining ( 0 , 0 , 0 ) (0,0,0) and ( 5 , 4 , 3 ) (5,4,3) be x = 5 t , y = 4 t , z = 3 t ( 0 ≤ t ≤ 1 ) x=5t,y=4t,z=3t (0\leq t\leq 1) . Then

F ⃗ = 5 i ^ + 16 t j ^ + 27 t 2 k ^ \vec F=5\hat i+16t\hat j+27t^2\hat k

d l ⃗ = ( 5 i ^ + 4 j ^ + 3 k ^ ) d t d\vec l=(5\hat i+4\hat j+3\hat k)dt

F ⃗ . d l ⃗ = ( 25 + 64 t + 81 t 2 ) d t \vec F.d\vec l=(25+64t+81t^2)dt

Therefore the required line integral is

∫ 0 1 ( 25 + 64 t + 81 t 2 ) d t = 25 + 32 + 27 \displaystyle \int_0^1 (25+64t+81t^2)dt=25+32+27

= 84 =\boxed {84} .

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