Evaluating Line Integrals for a Circle of Radius 3

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SUMMARY

The discussion focuses on evaluating the line integral of the function \((x^2+y^2)^2\) over a circle of radius 3 centered at the origin. The correct expression for the differential arc length \(ds\) is clarified as \(|r'(t)|dt\), rather than the incorrect formulation initially presented. The integral simplifies to a constant value of 81 when evaluated correctly, confirming the importance of accurately defining \(ds\) in line integrals.

PREREQUISITES
  • Understanding of line integrals in vector calculus
  • Familiarity with parametric equations for circles
  • Knowledge of differential arc length in polar coordinates
  • Basic integration techniques in calculus
NEXT STEPS
  • Study the derivation of arc length in polar coordinates
  • Learn about parametric equations for curves
  • Explore the application of line integrals in physics
  • Investigate the properties of conservative vector fields
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Students and educators in calculus, particularly those focusing on vector calculus and line integrals, as well as professionals applying these concepts in physics and engineering contexts.

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[SOLVED] Evaluating line integrals

Homework Statement


I am given a line integral:

\int (x^2+y^2)^2ds, where C is a circle of radius 3 with centre in (0;0).

Evaluate it.

The Attempt at a Solution


Ok, first I know (x^2+y^2)^2 = 81. So far, so good.

Then I know for an object in the xy-plane, ds = r*dr*d\theta. I just insert and use the correct limits, and do the integral?

Thanks in advance.
 
Physics news on Phys.org
No, I am totally wrong.

ds is _not_ what I wrote above, but ds = |r'(t)|dt.
 

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