Evaluate the integral with C be the boundary of the domain

[tavo0116]

Homework Statement

Let C be the boundary of the domain enclosed between y = x^2 and y = x. Assuming C is oriented counterclockwise. Evaluate the integral
∫c (6xy+e^(-x^2))dx

Homework Equations

I was thinking of using Green's Theorem. Would be the approach be correct?

The Attempt at a Solution

d/dy (6xy+e^(-x^2))dx = 6x, and integrate that I got 3x^2
What would be the limit? Would the problem be double integral since there is no "dy".
Would the limit be from 0 to 1 since y = x^2 is a parabola, and y = x is a diagonal line through the origin?
What about the dy? Would it need to be added? How?

 

[LCKurtz]

Homework Statement

Let C be the boundary of the domain enclosed between y = x^2 and y = x. Assuming C is oriented counterclockwise. Evaluate the integral
∫c (6xy+e^(-x^2))dx

Homework Equations

I was thinking of using Green's Theorem. Would be the approach be correct?

The Attempt at a Solution

d/dy (6xy+e^(-x^2))dx = 6x, and integrate that I got 3x^2
What would be the limit? Would the problem be double integral since there is no "dy".
Would the limit be from 0 to 1 since y = x^2 is a parabola, and y = x is a diagonal line through the origin?
What about the dy? Would it need to be added? How?

One side of Green's theorem is a double integral. If you were asked to find the area between those two curves, could you set up the double integral with limits for that? That is what you want, with the integrand being 6x.

 

[tavo0116]

But there is no "dy" , I still don't understand that.
Would the limit of x is from 0 to 1, and limit of y is from y to √y ?

 

[LCKurtz]

One side of Green's theorem is a double integral. If you were asked to find the area between those two curves, could you set up the double integral with limits for that? That is what you want, with the integrand being 6x.

But there is no "dy" , I still don't understand that.
Would the limit of x is from 0 to 1, and limit of y is from y to √y ?

You didn't answer my question.

Your original problem is$$
\int_C (6xy + e^{-x^2})dx + 0dy$$Does that help? Write down the other side of Green's theorem.

 

[tavo0116]

d/dx (0) - d/dy (6xy+e^(-x^2))
∫0 to 1 ∫0 to 1 (-6x)

I've tried to find the limits of x by setting up x = x^2, which then gives me x = 0, and x = 1, and for y, I use √y=y, which gives me y = 0, and y = 1

Would those be the right limits for the double integral?

 

[LCKurtz]

d/dx (0) - d/dy (6xy+e^(-x^2))
∫0 to 1 ∫0 to 1 (-6x)

I've tried to find the limits of x by setting up x = x^2, which then gives me x = 0, and x = 1, and for y, I use √y=y, which gives me y = 0, and y = 1

Would those be the right limits for the double integral?

No. 0 to 1 on both integrals describes a square. Your region isn't a square. Look in your calculus book for the formula for the area between two curves.