Final answer:
To evaluate the line integral ∫C y ds over the given curve, we need to parameterize the curve in terms of t and find the differential arc length ds. Then, we can substitute the values into the integral and evaluate it. The result is 168, so the correct answer is E. 168.
Step-by-step explanation:
To evaluate the line integral ∫C y ds over the curve given by x=√(3) t, y=8, z=√(6) t for 0 ≤ t ≤ 7, we first need to parameterize the curve in terms of t. From the given equations, we have x=√(3) t, y=8, and z=√(6) t. Now, we need to find ds, which represents the differential arc length along the curve. Since ds is equal to √((dx/dt)² + (dy/dt)² + (dz/dt)²) dt, we can substitute the values of dx/dt, dy/dt, and dz/dt into the formula to find ds. In this case, dx/dt = √(3), dy/dt = 0, and dz/dt = √(6). So, ds = √((√(3))² + 0² + (√(6))²) dt = √(3 + 6) dt = √9 dt = 3 dt.
Now, we can evaluate the line integral by substituting y = 8 and ds = 3 dt into the integral ∫C y ds. This gives us ∫07 (8)(3) dt = 24 ∫07 dt = 24(t)|07 = 24(7 - 0) = 168.
Therefore, the value of ∫C y ds over the given curve is 168. So, the correct answer is E. 168.