6. Suppose F⃗( x, y)= −yi⃗ + xj⃗ and C is the line segment from point P =( 5,0) to Q =( 0,3).( a) Find a vector parametric equation r⃗( t) for the line segment C so that points P and Q correspond to t = 0 and t = 1, respectively. r⃗( t)=( b) Using the parametrization in part( a), the line integral of F⃗ along C is ∫CF⃗ ⋅dr⃗ = ∫baF⃗( r⃗( t)) ⋅r⃗ ′( t) dt = ∫badt with limits of integration a = and b =( c) Evaluate the line integral in part( b).( d) What is the line integral of F⃗ around the clockwiseoriented triangle with corners at the origin, P, and Q? Hint: Sketch the vector field and the triangle.
7.
Use a CAS to calculate ∫� ⟨ ex−y, ex + y ⟩ ⋅ds to four decimal places, where � is the curve y = sinx for 0≤x≤π9, oriented from left to right.
Answer: 8.
Evaluate the line integral∫Cydx + xdywhereCis the parameterized pathx = t3, y = t2,2≤t≤5.
∫Cydx + xdy =
9. LetCbe the straight path from( 0,0) to( 5,5) and letF⃗ =( y−x−4) i⃗ +( sin( y−x) −4) j⃗.
( a) At each point of C, what angle does F⃗ make with a tangent vector to C? angle =( Give your answer in radians.)
( b) Find the magnitude ∥F⃗ ∥ at each point of C. ∥F⃗ ∥ =