Question

Evaluate the given integral along the indicated contour.

(8z − z) dz,
C
where C is
x = −t,
y = t2 + 2,
0 ≤ t ≤ 2

Answer 1

The question asks for the evaluation of a complex line integral along a given contour parameterized by x and y. The integral is simplified by expressing z and dz in terms of the parameter t, after which the integral is evaluated from 0 to 2.

Step-by-step explanation:

The question involves evaluating a complex line integral, specifically (8z - z) dz, along a given contour C, which is described parametrically by x = -t and y = t^2 + 2, where t varies from 0 to 2. To tackle this problem, we parameterize z using x and y where z = x + iy. We then express dz in terms of dt by differentiating x and y with respect to t, and substitute these expressions into the integral. Next, we evaluate the integral with respect to t from 0 to 2. This approach reduces the complex line integral into a more manageable form that depends only on one variable, t.