What is the Correct Answer for the Given Equation?

In summary, Green's theorem is a fundamental theorem in vector calculus that relates a line integral over a closed curve to a double integral over the region enclosed by the curve. It is significant because it simplifies complicated line integrals and has many applications in physics, engineering, and mathematics. It is a special case of Stokes' theorem and requires a good understanding of vector calculus, basic calculus, and multivariable calculus. Some applications include calculating work, fluid flow, and electric/magnetic fields in physics and engineering.
  • #1
denian
641
0
http://www.mrnerdy.com/forum_img/whichcorrect2.JPG

which one is correct?
[PLAIN]http://www.mrnerdy.com/forum_img/whichcorrect.JPG

i tried both, and both gives different answer.
one is 19/20 and another is 7/60
 
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  • #2
In the upper integral, you should go from [itex]-\sqrt{x}[/itex] to [itex]-x[/itex]. The other one is correct. Both integrals have the value -7/60. The integrand is negative in the region over which you're integrating ([itex]y\in [0,1][/itex]) so the answer should be negative. Lots of chances to mess up with minus signs here, so be careful...
 
  • #3
thank you very much, galileo
 

FAQ: What is the Correct Answer for the Given Equation?

What is Green's theorem?

Green's theorem is a fundamental theorem in vector calculus that relates the line integral of a 2-dimensional vector field over a closed curve to a double integral over the region enclosed by the curve.

What is the significance of Green's theorem?

Green's theorem is significant because it allows us to evaluate complicated line integrals by converting them into simpler double integrals, which are often easier to solve. It also provides a powerful tool for solving problems in physics and engineering that involve vector fields.

How is Green's theorem related to Stokes' theorem?

Green's theorem is a special case of Stokes' theorem, which is a more general version of the fundamental theorem of calculus. Stokes' theorem relates the line integral of a vector field over a surface to a surface integral of the curl of the vector field over the same surface.

What are the prerequisites for understanding Green's theorem?

To understand Green's theorem, one should have a good understanding of vector calculus, including vector operations such as dot and cross products, as well as line and surface integrals. Knowledge of basic calculus and multivariable calculus is also necessary.

What are some applications of Green's theorem?

Green's theorem has many applications in physics, engineering, and mathematics. It is commonly used to calculate work done by a force field, fluid flow, and electric or magnetic fields. It is also used in the study of fluid dynamics, electromagnetism, and other areas of physics and engineering.

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