What is the geometrical significance of definite integrals of vector functions?

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The discussion explores the geometrical significance of the definite integral of vector functions, particularly in the context of velocity. Integrating a velocity vector over time yields a displacement vector, indicating the particle's position change from t1 to t2. The resulting vector's direction reflects the particle's final position relative to its initial position. Participants note that while scalar functions have a clear area interpretation, vector functions represent distances traveled in three-dimensional space along the i, j, and k axes. The conversation highlights the complexity of finding a direct area interpretation for vector integrals compared to scalar integrals.
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What is the geometrical significance of the definite integral of a vector function if any?

e.g. if you integrate a vector function that gives the velocity of some particle between t1 and t2, the vector we get indicates the distance traveled in the i, j and k directions right? does the direction of this vector have any meaning? Also, is there a geometric interpretation of this value like area under the curve for the definite integral of a scalar function?
 
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specifically the integral of the velocity vector over time will give you the displacement vector, and the direction of this points in the direction of the particle's position at t2 relative to its position at t1.

I'm not sure what you want for a geometric interpretation. There are three curves and three areas- the distances traveled in each of the three directions of i, j and k.
 
Thanks. nah i was just wondering if there was somehow an area interpretation like with scalar functions
 

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