How to Integrate Along Curve C_1?

In summary, integrating along a curve C_1 allows for the calculation of a function's total value along a specific path, which is useful in fields such as physics and engineering. The main difference between integrating along a curve and a straight line is the number of points used in the calculation. The limits of integration for a curve C_1 are determined by the start and end points of the curve, and integration can be done in any direction as long as the path is continuous and limits are correctly chosen. Some applications of integrating along a curve C_1 include calculating work, fluid flow, and analyzing complex systems.
  • #1
Ted123
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Homework Statement



[PLAIN]http://img580.imageshack.us/img580/9506/linegu.jpg

The Attempt at a Solution



Parametrising the curve [itex]C_1[/itex] as follows:

[itex]{\bf p}(t) = (1-t)(1,4) + t(3,3) = (1+2t,4-t)[/itex]


[itex]{\bf p}'(t) = (2,-1)[/itex]

So [itex]\int_{C_1} = \bigg\{ \sqrt{\frac{y}{x}} \frac{dx}{dt} + \sqrt{\frac{x}{y}} \frac{dy}{dt} \bigg\} \;dt = \int^1_0 \bigg\{ 2\sqrt{\frac{4-t}{1+2t}} - \sqrt{\frac{1+2t}{4-t}} \bigg\} \;dt[/itex]

How do I do this integration?
 
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  • #2
i would attempt to rationalise the integrands
 

FAQ: How to Integrate Along Curve C_1?

What is the purpose of integrating along a curve C_1?

The purpose of integrating along a curve C_1 is to calculate the total value of a function along a specific path. This is useful in many scientific fields, such as physics and engineering, where the path of an object or phenomenon may not be a straight line.

What is the difference between integrating along a curve and a straight line?

The main difference is that integrating along a curve involves calculating the value of a function at multiple points along a curved path, while integrating along a straight line involves calculating the value of a function at only two points (the start and end points).

How do I choose the limits of integration for a curve C_1?

The limits of integration for a curve C_1 are determined by the start and end points of the curve. This can be done by plotting the curve and identifying the points where it begins and ends. Alternatively, if the curve is described by a mathematical equation, the limits can be calculated using that equation.

Can I integrate along a curve C_1 in any direction?

Yes, you can integrate along a curve C_1 in any direction as long as the path remains continuous and the limits of integration are correctly chosen. However, the direction of integration may affect the final value of the integral, so it is important to consider the direction carefully.

What are some applications of integrating along a curve C_1?

Integrating along a curve C_1 has many practical applications in physics, engineering, and other scientific fields. For example, it can be used to calculate the work done by a force on an object moving along a curved path, or the total flow of a fluid through a curved pipe. It can also be used in mathematical modeling to analyze the behavior of complex systems.

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