Question

Evaluate the line integral, where C is the given curve, I=∫c x² zds, C is the line segment from (0,6,−1) to (4,1,5).

A. I = 56/3 √77
B. I = 96/3 √77
C. I = 5/3 √77
D. I = 1
E. None

Answer 1

To evaluate the line integral, we need to parametrize the given line segment and then integrate. The line integral of x²z over the line segment from (0,6,-1) to (4,1,5) is 56/3 √77.

Step-by-step explanation:

To evaluate the line integral, we need to parametrize the given line segment. Let's parameterize the line segment from (0,6,-1) to (4,1,5) using the parameter t, where 0 ≤ t ≤ 1.

The x-coordinate can be expressed as x = 0 + 4t = 4t.

The y-coordinate can be expressed as y = 6 - 5t.

The z-coordinate can be expressed as z = -1 + 6t.

The line integral becomes: I = ∫C x²z ds = ∫01 (4t)²(-1 + 6t) ||r'(t)|| dt.

Now, we need to find ||r'(t)||, which is the magnitude of the derivative of r(t).

||r'(t)|| = ||<4, -5, 6>|| = √(4² + (-5)² + 6²) = √(16 + 25 + 36) = √77.

Substituting ||r'(t)|| and integrating, we get I = ∫01 (16t²(-1 + 6t))/√77 dt = 56/3 √77.