Compute line integral for vector function along the curve

Question

The question is to compute $\int_{C}{\vec{F} \cdot d\vec{R}}$ from the origin to (1, 1, 1) along the curve $x=y=z$ and along the curve $y=x^2$, $z=x^3$.

Where $ \vec{F}=(2xyz^3,-x^2z^3-2y,3x^2yz^2$.

I separate the line integral into to parts, the first part is the curve $x=y=z$. The parametric curve would be $ \vec{R_1}=(t,t,t)$ where $ 0\leq t \leq1$.

The second curve is the intersection of the curves $y=x^2$, $z=x^3$. The parameterized curve is $ \vec{R_2}=(t,t^2,t^3)$.

But now, I am stuck on the limits for t for $ \vec{R_2}$.

Any help please? Thank you!

Answer 1

I' ll do it for the first

we have:

$d\vec{R}=(dx,dy,dz)$

and

$x=y=z$.

so

$dx=dy=dz$

and

$$\vec{F}=(2x^5,-x^5-2x,3x^5)$$.

thus

$\vec{F}.d\vec{R}=$

$(2x^5-x^5-2x+3x^5)dx=$

$(4x^5-2x)dx$.

the integral is

$$\int_0^1(4x^5-2x)dx=$$

$$\frac{2}{3}-1=-\frac{1}{3}$$.

FOR THE SECOND, you have $dy=2xdx$ and $dz=3x^2dx$. i'm sure you could continue.