Find the line integral with respect to arc length ∫C(7x+6y)ds, where C is the line segment in the xy-plane with endpoints P=(8,0) and Q=(0,4).

Question

 Find a vector parametric equation r⃗ (t)r→(t) for the line segment C so that points P and Q correspond to t=0 and t=1, respectively.

r→(t)= 

(b) Using the parametrization in part (a), the line integral with respect to arc length is

∫ba∫C(7x+6y)ds=∫ab  dt

with limits of integration a=  and b=

(c) Evaluate the line integral with respect to arc length in part (b).

∫C(7x+6y)ds=

Answer 1

(a) Parametric equation: r = <8, 0> + t<0 - 8, 4 - 0> = <8, 0> + t<- 8, 4>; 0 ≤ t ≤ 1.

x = 8 - 8t, y = 4t This is parametric equation of C

(b,c) ∫_C(7x + 8y)ds = ∫₀¹(56 - 56t + 32t)√(64 + 16)dt = (56t -12t²)·√80₀¹ = 44·4√5 = 176√5