Question

Evaluate the line integral, where C is the given curve. z2 dx x2 dy y2 dz, C C is the line segment from (1, 0, 0) to (5, 1, 2)

Answer 1

I assume the integral is supposed to be

`\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz`

Parameterize C by

`(x(t),y(t),z(t)) = (1-t)(1,0,0) + t(5,1,2) = (1+4t, t, 2t)`

with 0 ≤ t ≤ 1, and dx = 4 dt, dy = dt, and dz = 2 dt. Then the line integral is

`\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = \int_0^1 (2t)^2(4\,dt) + (1+4t)^2\,dt + t^2(2\,dt)`

`\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = \int_0^1(1+8t+34t^2)\,dt`

`\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = 1+4+\frac{34}3 = \boxed{\frac{49}3}`