Question

Let (C) be the curve defined by (y = x^2 - 2x + 3). Determine the line integral of (y) with respect to arc length over the line segment from (1, 3) to (4, 5).

a) (13)
b) (18)
c) (21)
d) (24)

Answer 1

The line integral of function y over the given line segment is calculated by integrating the product of y and the differential arc length ds, which depends on the derivative dy/dx, from x=1 to x=4.

Step-by-step explanation:

The student has asked to determine the line integral of a function y with respect to arc length over a specific line segment. This line segment runs from the point (1, 3) to the point (4, 5) along the curve y = x^2 - 2x + 3. To find this, we will use the formula for the line integral with respect to arc length, which can be expressed in terms of a single variable. Using x as our parameter, we first identify the arc length differential ds, which is obtained from the formula ds = √(1 + (dy/dx)^2)dx. Then, we integrate the function y multiplied by this differential over the interval from x=1 to x=4.

The derivative of y with respect to x is dy/dx = 2x - 2. Substituting this into the formula for ds, we obtain ds = √(1 + (2x - 2)^2)dx. The line integral in question is then the integral from x=1 to x=4 of the product of y and ds, which translates to ∫(y √(1 + (dy/dx)^2)) dx from x=1 to x=4. After performing the integration and evaluating at the bounds, we would obtain the value of the line integral.