Question

Find the line integral with respect to arc length ∫¶(4x 9y)X‘X , where ¶ is the line segment in the xy-plane with endpoints P=(5,0) and X„=(0,7). (a) Find a vector parametric equation XŸƒ—(X¡) for the line segment ¶ so that points P and X„ correspond to X¡=0 and X¡=1, respectively.

Answer 1

The parametric equation of the line segment connecting P=(5,0) to Q=(0,7) is X(t) = (5 - 5t, 7t), and is essential for evaluating the line integral of the vector field (4x, 9y) along this path.

Step-by-step explanation:

To find the line integral concerning the arc length of the vector field (4x, 9y) along the line segment from P=(5,0) to Q=(0,7), we first need to find a parametric equation for this line segment. The parametric equation of a line is given by X(t) = P + t(Q - P), where t varies from 0 to 1. For our line segment, P = (5,0) and Q = (0,7), we have:

X(t) = (1-t)(5,0) + t(0,7)

which simplifies to:

X(t) = (5 - 5t, 7t)

This vector equation represents the straight line segment connecting P and Q with a parameter t varying from 0 to 1. To evaluate the line integral over this path, we would next find the derivative of X(t) concerning t, X'(t), then substitute X(t) and X'(t) into the integral expression and integrate concerning t from 0 to 1.