Vector Calc 8-11-2020 (Part 2)

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 ) = ( − y , x ) \vec{F} = (F_x, F_y) = (-y , x) 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 661.47.

Steven Chase by Steven Chase , 37, USA

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2 solutions

F ⃗ . d l ⃗ = x d y − y d x = θ 2 d θ \vec F.d\vec l=xdy-ydx=\theta^2d\theta

So the required integral is

∫ 0 4 π θ 2 d θ = 64 π 3 3 \displaystyle \int_0^{4π} \theta^2d\theta=\dfrac {64π^3}{3}

≈ 661.467 \approx \boxed {661.467} .

2 Helpful 2 Interesting 2 Brilliant 0 Confused

@Foolish Learner Nice, upvoted. I don't know why I don't able to think this.

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

Nice problem

1 Helpful 1 Interesting 2 Brilliant 0 Confused

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