Calculus line integrals

Question

Evaluate  where , and  is the circle of radius  centered at the origin parametrized in the counterclockwise direction.

Answer 1

∫_CF • r = ∫^t=2pi_t=0 F ( r (t ) ) • r' ( t ) d t where r (t ) = < 4 cos t , 4 sin t >

∫_CF • r = ∫^t=2pi_t=0 F ( r (t ) ) • r' ( t ) d t = ∫^t=2pi_t=0 F < 4 cos t - 4 sin t , 4 cos t + 8 sin t > • < -4 sin t , 4 cos t > dt

= ∫^t=2pi_t=0 [-4sint ( 4 cos t - 4 sin t) +4 cost[4cos t + 8 sin t d t

= ∫^t=2pi_t=0 [-16sint cos t + 16 sin ²t +16 cos²t+ 32 sin t cos t ]d t

∫^t=2pi_t=0 [16sint cos t + 16]d t ∫^t=2pi_t=0 [16sint cos t ] d t +∫^t=2pi_t=0 [ 16]d t = 32π