Find \displaystyle \int_C \vec F \cdot d\vec r for the given \vec F and C. \vec F = -y\vec i +...

Question:

Find {eq}\displaystyle \int_C \vec F \cdot d\vec r {/eq} for the given {eq}\vec F {/eq} and {eq}C {/eq}.

{eq}\vec F = -y\vec i + x\vec j + 6\vec k {/eq} and {eq}C {/eq} is the helix {eq}x = \cos t, y = \sin t, z = t {/eq}, for {eq}0 \leq t \leq 3\pi {/eq}.

Line Integral:

To solve the line integral let us recall the following formula {eq}\displaystyle \int _C F\cdot dr=\int _C F\left ( r(t) \right )\cdot r'(t)dt {/eq} where {eq}r(t) {/eq} is the vector equation of the curve and {eq}F {/eq} is the vector field.

Answer and Explanation: 1

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From the given above,

{eq}\displaystyle r(t)=\cos t\:i+\sin t\:j+t\:k {/eq}

{eq}\displaystyle r'(t)=-\sin t\:i+\cos \:j+k {/eq}

Thus,

{eq}\di...

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Line Integrals: How to Integrate Functions Over Paths

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Chapter 15 / Lesson 2
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Line integrals are any integral of a function that can be defined along a given curve in a three-dimensional space. Learn the process of line integration and how they can be used to map paths using parametrizations.


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