What is the Line Integral of xydx+4ydy along a Curve from (1,2) to (3,5)?

In summary, the given problem involves finding the integral of xydx+4ydy along the curve C, which is made up of two line segments parallel to the coordinate axes. One segment, c1, goes from (1,2) to (3,2) and the other, c2, goes from (3,2) to (3,5). The correct solution involves calculating the integral for each segment separately, taking into account the constant values of y for c1 and x for c2. The integration can be simplified by substituting specific values for x and y, as done in the solution provided.
  • #1
qq545282501
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1

Homework Statement


[tex] \int xydx+ 4ydy[/tex]
where C is the curve from (1,2) to (3,5) made up of the twoline segments parallel to the coordinate axes.
[tex]c_1:(1,2)\rightarrow(3,2)[/tex]
[tex]c_2:(3,2)\rightarrow(3,5)[/tex]

Homework Equations

The Attempt at a Solution


i got c2 correct, y=2+3t, and x = 0, for t goes from 0 to 1.
but i got c1 wrong, for c1, i see only x is changing, x=1+2t. so x'(t)= 2. if y value is not changing, it means that dy=0, my professor had y=1, by setting 2xy=2x, i guess 2xy is the first half of the initial integral by replacing dx with 2, but i don't understand what is 2x on the right side of the equation.
 
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  • #2
Hi qq:

I don't understand why you introduce the t variable unless it is that you are used to doing that in general for arbitrary curves. I also don't understand why you use t=0 and t= 1 as the limits. For this particular problem I suggest you write down the integrals with the limits specified for both c1 and c2. Since y is constant for c1 and x is constant for c2, these two integrals should be easy to integrate.

Hope t his helps.

Regards,
Buzz
 
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  • #3
##\int xy\, dx## with y constant is ## y \int x\, dx##. For the first leg that means the same as xy = 2x and you calculate ##2\int_1^3 x\, dx##.

If you want to end up with an ##\int_0^1## ( a sort of parametrization that isn't really necessary here, as BB explains), you substitute t = (x-1)/2 so dt = dx/2 to get ##2 \int_0^1 2(1+2t) \, dt ##

same result :rolleyes:

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FAQ: What is the Line Integral of xydx+4ydy along a Curve from (1,2) to (3,5)?

1. What is a line integral?

A line integral is a mathematical concept used in multivariable calculus to measure the total value of a function along a curve in a vector field. It is represented by the symbol ∫ and is typically used to calculate work, mass, or other physical quantities.

2. How do you evaluate a line integral?

To evaluate a line integral, you first need to determine the parametric equations for the curve in the vector field. Then, you can use the formula ∫F(x,y)ds = ∫F(x(t),y(t))√(x'(t)^2 + y'(t)^2)dt, where F(x,y) is the function being integrated, x(t) and y(t) are the parametric equations, and ds is the infinitesimal length of the curve. You can then use integration techniques to solve the integral.

3. What is the difference between a line integral and a surface integral?

A line integral is used to measure the value of a function along a curve, while a surface integral is used to measure the value of a function over a surface. Line integrals are one-dimensional, while surface integrals are two-dimensional. Additionally, surface integrals involve a double integral, while line integrals only involve a single integral.

4. When is a line integral equal to zero?

A line integral is equal to zero when the function being integrated is equal to zero at every point on the curve. This means that the curve does not contribute to the overall value of the integral, and the integral becomes trivial. Alternatively, if the curve is closed and the function being integrated is conservative, then the line integral will also be equal to zero.

5. What are some real-world applications of line integrals?

Line integrals have various real-world applications in physics, engineering, and economics. For example, they can be used to calculate the work done by a force along a path, the mass of a wire or rope, or the flow rate of a fluid through a pipe. They can also be used to measure the circulation of a vector field or the potential energy of an electric field.

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