Evaluating Line Integrals for a Circle of Radius 3

In summary, a line integral is a type of integral used to calculate the value of a function along a given curve or path. It is necessary in physics, engineering, and mathematics for solving various problems and determining physical quantities. To evaluate a line integral, the curve must first be parameterized and then integrated with respect to a variable. Line integrals differ from double integrals in the dimensionality of the region being integrated over. They have applications in fields such as physics, engineering, mathematics, computer graphics, and image processing.
  • #1
Niles
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[SOLVED] Evaluating line integrals

Homework Statement


I am given a line integral:

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

Evaluate it.

The Attempt at a Solution


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

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

Thanks in advance.
 
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  • #2
No, I am totally wrong.

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

1. What is a line integral?

A line integral is a type of integral used in mathematics and physics to calculate the value of a function along a given curve or path. It takes into account the direction and length of the curve to determine the total value of the function.

2. Why do we need to evaluate line integrals?

Line integrals are used to solve various problems in physics, engineering, and mathematics. They allow us to calculate physical quantities such as work, mass, and charge distribution along a given path or curve. They are also useful in evaluating vector fields and determining potential functions.

3. How do you evaluate a line integral?

To evaluate a line integral, you must first parameterize the given curve or path. This involves expressing the coordinates of the curve in terms of a single variable. Then, you can plug the parameterized equation into the integrand and integrate with respect to the variable. The result will be a single numerical value.

4. What is the difference between a line integral and a double integral?

While both line integrals and double integrals involve finding the area under a curve, they differ in the dimensionality of the region being integrated over. Line integrals are used to integrate over curves in two or three dimensions, while double integrals integrate over regions in two-dimensional space.

5. In what fields are line integrals used?

Line integrals have applications in various fields, including physics, engineering, and mathematics. They are commonly used to calculate work, electric and magnetic fields, fluid flow, and probability. They are also used in computer graphics and image processing to calculate the brightness and color of pixels along a given path.

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