Area enclosed by an elliptic spiral curve integral

[gipc]

I need help with the following Integral
http://img258.imageshack.us/img258/469/55758717.jpg

When C is the elliptic spiral that is given in the parametric form
http://img138.imageshack.us/img138/8549/48152171.jpg

 

[lanedance]

so have you tried anything?

 

[lanedance]

here's how you can write it in tex
$$\int_C y dx-x dy +dz$$

note you can re-write as a dot product, just for clarity
$$\int_C (y,-x,1) \bullet \vec{dx}$$

to solve, you just need to re-write the integral so you're only integrating over t

 

[gipc]

here's how you can write it in tex
$$\int_C y dx-x dy +dz$$

note you can re-write as a dot product, just for clarity
$$\int_C (y,-x,1) \bullet \vec{dx}$$

to solve, you just need to re-write the integral so you're only integrating over t

So the integral becomes $$\int_C$$ [b*sint*dx-a*cost*dy+dz]
Which equals $$\int_C$$ -a*b*cost*sint*dt-a*b*cost*cost*dt+c*dt ? And now should I separate the integrals?

Is that all I have to solve or is something wrong?

 

[LCKurtz]

here's how you can write it in tex
$$\int_C y dx-x dy +dz$$

note you can re-write as a dot product, just for clarity
$$\int_C (y,-x,1) \bullet \vec{dx}$$

to solve, you just need to re-write the integral so you're only integrating over t

Or, a little neater write the integrand

$$\langle y, -x, 1\rangle \cdot \langle dx,dy,dz\rangle$$

 

[gipc]

So the integral becomes $$\int_C$$ [b*sint*dx-a*cost*dy+dz]
Which equals $$\int_C$$ -a*b*cost*sint*dt-a*b*cost*cost*dt+c*dt ? And now should I separate the integrals?

Is that all I have to solve or is something wrong?

so... is it okay to follow this steps?

 

[HallsofIvy]

So the integral becomes $$\int_C$$ [b*sint*dx-a*cost*dy+dz]

Yes, that is correct.

Which equals $$\int_C$$ -a*b*cost*sint*dt-a*b*cost*cost*dt+c*dt ?

No, that is not correct. You are given that x= a cos t so dx= -a sin(t). b sin t dx= (b sin t)(-a sin t dt)= -ab sin^2 t dt. y= b sin t so dy= b cos t dt. -a cos t dy= (-a cos t)(b cos t dt)= -ab cos^2 t dt. z= ct so dz= ct as you have.

And now should I separate the integrals?

Is that all I have to solve or is something wrong?