Vector Calc 8-11-2020

Calculus Level 3

Consider the curve:

x = θ cos ⁡ θ y = θ sin ⁡ θ x = \theta \cos \theta \\ y = \theta \sin \theta

Determine the value of the line integral of the vector field F ⃗ = ( F x , F y ) = ( 2 x , 2 y ) \vec{F} = (F_x, F_y) = (2x , 2y) over the curve from θ = 0 \theta = 0 to θ = 4 π \theta = 4 \pi .

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


The answer is 157.91.

Steven Chase by Steven Chase , 37, USA

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2 solutions

Steven Chase
Aug 11, 2020

@Lil Doug has shown how to evaluate the line integral directly. Another way is to note that the vector field is the gradient of the potential function U = x 2 + y 2 = r 2 U = x^2 + y^2 = r^2 .

∫ F ⃗ ⋅ d ℓ ⃗ = U 2 − U 1 = ( 4 π ) 2 − 0 2 = 16 π 2 \int \vec{F} \cdot \vec{d \ell} = U_2 - U_1 = (4 \pi)^2 - 0^2 = 16 \pi^2

3 Helpful 3 Interesting 3 Brilliant 0 Confused

@Steven Chase please check last 5 hr notification.

Talulah Riley - 10 months ago
Talulah Riley
Aug 11, 2020

Nice problem.

3 Helpful 2 Interesting 3 Brilliant 0 Confused

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