A(0,10, 0), B($3 \sqrt 10, 0, \sqrt10$), C(2, 8, 0) with $\int_\Gamma (x^2 + 5y + 7z)dx + (y^2+z+5x)dy + (z^2+7x + y)dz)$
$\Gamma$ is made from the following segments:
A->B {$ (x,y,z)∈ \Bbb R^3 / x^2+y^2+z^2=100,x-3z=0, x≥0, y≥0$}
[BA] line
From what I understand the A B portion is a quarter of a sphere with the radius of 10 and a plane that goes through the origin and cuts the quarter sphere? So I think I should apply Stokes' Theorem on the first part then I substract from that the second part somehow
Sorry for my bad english and formatting. Thank you for the future insight
Please note that the vector field $\vec F = (x^2 + 5y + 7z, y^2+z+5x, z^2+7x + y)$ is a conservative vector field and its potential function is
$\vec F = \nabla f(x,y,z)$ where $\displaystyle f(x,y,z) = \frac{x^3+y^3+z^3}{3} + 5xy + yz + 7zx$.
So the line integral is path independent and it should simply be
$I = f(c) - f(a)$ where $a$ is the starting point and $c$ is the end point.
Can you take it from here?