Integral path between 2 points

Question

so I need your help calculating the next inegral:

Calculate the integral $$\int(10x^4-2xy^3)dx -3x^2y^2dy$$ at the path $$x^4-6xy^3=4y^2$$

between the points $O(0,0)$ to $A(2,1)$

please explain me how to do it... thanks!

what i tried to do:

$$\int(10x^4-2xy^3)dx -3x^2y^2dy=\int_0^2(10x^4+\frac{4-x^3}{3})dx$$ not sure if its correct ...

Answer 1

First, note that the derivative of the coefficient of "dx",$10x^4- 2xy^3$, with respect to y, is $-6xy^2$ and that the derivative of the coefficient of "dy", $-3x^2y^2$ is with respect to x also $-6xy^2$. That tells us that the integral is independent of the path. One method of doing this would be to choose some simple path, say the straight line between (0, 0) and (2, 1) or perhaps the "broken line" from (0, 0) to (2, 0) and then from (2, 0) to (2, 1).

But a more fundamental method is to use the "fundamental theorem of calculus"- find a function, f(x, y), such that $df= f_x dx+ f_ydy= (10x^4- 2xy^3)dy- 3x^2y^3dy$. That means we must have $f_x= 10x^4- 2xy^3$ and, since that partial derivative is just the derivative with respect to x, treating y as a constant, we take the antiderivative treating y as a constant: $f(x,y)= 2x^5- x^2y^3+ u(y)$. Notice the "u(y)". That is the "constant of integration" but, since we are treating y as a constant, it may, in fact be a function of y. Differentiating that function with respect to y, $f_y= -3x^2y^2+ u'(y)$ and that must be equal to $-3x^2y^2$. So $-3x^2y^2+ u'= -3x^2y^2$ and u really is a constant. We have $f(x, y)= 2x^5- x^2y^3+ C$. Evaluate that at (2, 1) and (0, 0) and subtract.

This is all pretty basic for path integrals. I am surprised that you would be given a problem like this if you had not seen that before.

Answer 2

In a question like this, you first try the fundamental theorem for line integrals and then, if that fails, attempt harder techniques.

Summarizing from wikipedia: If $f(x,y)$ is a differentiable function on $\mathbb{R}^2$ and $\gamma$ a path from $(a,b)$ to $(c,d)$, then $$ f(c,d)-f(a,b)=\int_\gamma f_x(x,y)dx+f_y(x,y)dy. $$ (I am being very lazy with the conditions here, one can / should be more precise).

In this case, if you can find a "nice" function $f$ so that $f_x=10x^4-2xy^3$ and $f_y=-3x^2y^2$, then the path doesn't matter and you can just plug in the endpoints.

I encourage you to review the fundamental theorem for line integrals to make sure that you understand the conditions and when it can be applied.