Evaluating Line Integrals Along a Curve

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SUMMARY

This discussion focuses on evaluating line integrals along a specified curve using the integral expression \(\int(x*z*y)dx - (x-y)dy + (x^3)dz\) from the point (1,0,0) to (1,0,2π along the curve defined by \((x,y,z)=(\cos(t),\sin(t),t)\). The process involves splitting the line integral into components and utilizing the dot product to simplify calculations. The correct limits for the integral are from \(t=0\) to \(t=2\pi\), emphasizing the importance of evaluating the definite integral with respect to \(t\) for accurate results.

PREREQUISITES
  • Understanding of line integrals in vector calculus
  • Familiarity with parametric equations for curves
  • Knowledge of dot products in vector analysis
  • Ability to perform definite integrals
NEXT STEPS
  • Study the properties of line integrals in vector fields
  • Learn how to convert parametric equations into integral forms
  • Explore techniques for evaluating definite integrals in calculus
  • Investigate applications of line integrals in physics and engineering
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Students and professionals in mathematics, physics, and engineering who are working with vector calculus and need to evaluate line integrals along curves.

bfr
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If a question says something like: "evaluate \int(x*z*y)dx - (x-y)dy + (x^3)dz from (1,0,0,) to (1,0,2pi) along the curve (x,y,z)=(cos(t),sin(t),t)" or something like that, this is just basically splitting up a line integral? In my example, it would be the same as: \intcos(t)*t*sin(t)) * (-sin(t)) dt - \int(cos(t)-sin(t))*cos(t) dt ... etc. , which is just: \int<(cos(t)*t*y),-(cos(t)-y),(cos(t)^3)> dot <-sin(t),cos(t),1> dt from t=0 to t=1 ("dot" represents a dot product).
 
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bfr said:
If a question says something like: "evaluate \int(x*z*y)dx - (x-y)dy + (x^3)dz from (1,0,0,) to (1,0,2pi) along the curve (x,y,z)=(cos(t),sin(t),t)" or something like that, this is just basically splitting up a line integral? In my example, it would be the same as: \intcos(t)*t*sin(t)) * (-sin(t)) dt - \int(cos(t)-sin(t))*cos(t) dt ... etc. , which is just: \int<(cos(t)*t*y),-(cos(t)-y),(cos(t)^3)> dot <-sin(t),cos(t),1> dt from t=0 to t=1 ("dot" represents a dot product).
Pretty much, though the limits would be from t=0 to t=2pi, and it's the second form of the integral (i.e. the expanded dot product) that is useful to calculate (by evaluating the definite integral with respect to t).
 
OK, thanks.

And, er, yeah, I meant from t=0 to t=2pi.
 

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