Integral of F on Curve C: Evaluate

  • Thread starter Tom McCurdy
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In summary, the problem involves evaluating the integral of Fdr for F = (2xy^4)i+(2x^2y^3)j on the curve C, which is made up of the x-axis from x=0 to x=1, the parabola y=1-x^2 up to the y-axis, and the y-axis down to the origin. The solution involves using Green's Theorem and solving a double integral, which results in the answer of -1/10.
  • #1
Tom McCurdy
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The problem:

Evaluate the integral
[tex] \oint_{c} Fdr [/tex] for [tex] F = (2xy^4)i+(2x^2y^3)j [/tex] on the curve C consisting of the x-axis from x=0 to x=1, the parabola y=1-x^2 up to the y-axis, and the y-axis down to the origin

Here is what I triedF(x,y)=[tex]<2xy^4,2x^2y^3> [/tex][tex] \int\int_{D} =[(2x^2y^3)*\frac{\partial}{\partial x}-(2xy^4)\frac{\partial}{\partial y}] dA [/tex]= [tex] \int_{0}^{1} \int_{1-x^2}^{0} [4xy^3-8xy^3] dy dx [/tex]=1/10
 
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  • #2
Ok I have redone this problem... multiple times... and now I am getting

-1/10

can someone confirm this for me, I am out of tries on my submission so I need to be sure when I submit this problem
 
  • #3
alright I solved it nevermind
 
  • #4
I am so confused by the double integral
 

Related to Integral of F on Curve C: Evaluate

1. What is the definition of the integral of a function on a curve?

The integral of a function on a curve is a mathematical concept that represents the area under the curve. It is a measure of the accumulation of values of the function as it varies along the curve.

2. How do you evaluate the integral of a function on a curve?

To evaluate the integral of a function on a curve, you need to use a specific method called integration. This involves breaking down the curve into smaller segments and approximating the area under each segment. The sum of these approximations gives the value of the integral.

3. What is the relationship between the integral of a function on a curve and its antiderivative?

The integral of a function on a curve is closely related to its antiderivative. In fact, the antiderivative is the opposite of the derivative, and the derivative is the rate of change of the function. Therefore, the integral represents the total value of the function over a given interval.

4. Can the integral of a function on a curve be negative?

Yes, the integral of a function on a curve can be negative. This occurs when the function is below the x-axis, meaning that the area under the curve is below the x-axis and therefore has a negative value.

5. What are some real-life applications of finding the integral of a function on a curve?

The integral of a function on a curve has many real-life applications, including calculating the work done by a variable force, finding the distance traveled by an accelerating object, and determining the average value of a function. It is also used in areas such as physics, engineering, economics, and statistics.

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