Tuesday, March 4 ∗∗ Arclength and line integrals.1. Consider the curve C given by x2+ y2= 4.(a) Draw a sketch of this curve.··2·1·2·-1·-2·-2·-11(b) Find a parameterization of this curve.x(t ) = 2cos t, y(t) = 2sin t, 0 ≤ t ≤ 2π.2. Compute the arc length of the curve C. Using the above parameterization,L =Z2π0qx0(t )2+ y0(t )2d tx0(t ) = −2 sin t , y0(t ) = 2 cos t,px0(t )2+ y0(t )2= 2, soL =Z2π0qx0(t )2+ y0(t )2d t =Z2π02d t = 4π3. Let f (x, y) = 2x y. ComputeRCf d s. Using the above parameterization, we havef (x(t ), y(t)) = 2cos t sin t = cos2t.We haved s =qx0(t )2+ y0(t )2d t = 2d t ,soZCf d s =Z2π02cos2td t .Making the u-substitution u = 2t, we have du = 2d t, and so our integral isZ4π0cos td t = − sin t|4π0= (−0) − (−0) = 0.4. ComputeRCx2d x,RCy2d y. Using the above parameterization d x = −2sin td t, d y = 2 cos t d t . SoZCx2d x =Z2π04cos2t(−2sin t)d t = 8Z2π0cos2t(− sin t)d tMaking the u-substitution u = cos t, du = (−sin t)d t, our integral becomes8Z00u2du = 0.Similarly,ZCy2d y =Z2π04sin2t(2cos t)d t = 8Z2π0sin2t cos td t.Making the u-substitution u = sin t, du = cos td t, our integral becomes8Z00u2du =
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