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 ∥ =