Calculating Line Integral of (x^3-y^3)dx +(x^3+y^3)dy

In summary, the line integral over the region bounded by x^2+y^2=1 and x^2+y^2=9 is 120*Pi + 3*Pi/2. However, the question is unclear as it does not specify the direction for each part of the contour. This could affect the final answer.
  • #1
scarebyte
11
0
1.find the line integral of
(x^3-y^3)dx +(x^3+y^3)dy over r, where r is the boundary of the
region limited by x^2+y^2=1 and x^2+y^2=9





Homework Equations





3.
i found that the
line integral over the curve x^2+y^2=1 is 3*Pi/2
and the double integral of the region limited by x^2+y^2=1 and x^2+y^2=9 is 120*Pi
so the answer would be 120*Pi + 3*Pi/2 ?


 
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  • #2
The question is not very clear. The region between x^2+y^2=1 and x^2+y^2=9 consists of two curves that aren't connected. A proper question would specify the direction for each part of the contour. 120*pi is correct if you make a certain assumption about the direction of each contour. Whoever is giving you these questions should really work on phrasing them better.
 

FAQ: Calculating Line Integral of (x^3-y^3)dx +(x^3+y^3)dy

What is a line integral?

A line integral is a type of integral used in multivariable calculus to calculate the area under a curve or the work done along a path in a vector field. It is a way to measure the total change of a function along a given path.

What is the formula for calculating a line integral?

The formula for calculating a line integral is ∫(f(x,y)dx + g(x,y)dy), where f(x,y) and g(x,y) are the components of the vector field along the path and dx and dy represent the infinitesimal changes in the x and y directions, respectively.

How do you calculate a line integral of a specific function?

To calculate a line integral of a specific function, you first need to parameterize the path along which the integral will be evaluated. Then, substitute the parameterized values into the formula and integrate over the given limits of the path.

What is the significance of the line integral of (x^3-y^3)dx + (x^3+y^3)dy?

The line integral of (x^3-y^3)dx + (x^3+y^3)dy is significant because it represents the work done along a path in a vector field with components x^3-y^3 and x^3+y^3. It can also be interpreted as the net change of a function along the given path.

How is the line integral of (x^3-y^3)dx + (x^3+y^3)dy related to physical applications?

The line integral of (x^3-y^3)dx + (x^3+y^3)dy has various physical applications, such as calculating the work done in a force field, the circulation of a fluid, or the flux of a vector field. It is also used in engineering and physics to analyze the behavior of electric and magnetic fields.

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