676CHAPTER 1036.Letbe the region enclosed by the loop of the curve inExample 1.(a)Find the area of(b) Ifis rotated about the-axis, find the volume of theresulting solid.(c)Find the centroid ofShow that the total length of the ellipse,, isxxtetytet0t21e2sin2dwhereis the eccentricity of the ellipse, where.54.Find the total length of the astroid,where..37.,,,,40.,41–44Find the exact length of the curve.41.,42.,43.,44.,tstytst0t10t1y42t3x13t20t3y52txetet0t1ytcostxtsin0ty3 sintsin 3tx3 costcos 3,,,,,;45–46Graph the curve and find its length.,,;47.Graph the curve,and find itslength correct to four decimal places.48.Find the length of the loop of the curve.49.Use Simpson’s Rule withto estimate the length of thecurve,,3t2,.50.In Exercise 43 in Section 10.1 you were asked to derive theparametric equations,for thecurve called the witch of Maria Agnesi. Use Simpson’s Rulewithto estimate the length of the arc of this curvegiven by51–52Find the distance traveled by a particle with positionasvaries in the given time interval. Compare with the length ofthe curve.,t.,52.,t4ycostxcos2xasinab0ybcos,53.Show that the total length of the ellipse,, isxxtetytet0t21e2sin2dwhereis the eccentricity of the ellipse, where.54.Find the total length of the astroid,where,4xtstytst0t10t1y42t3x13t20t3y52txetet0t1ytcostxtsin0ty3 sintsin 3tx3 costcos 3ttt1ty3t2x,ty0t4ycostxcos2xasinab0ybcostL4ay20s1e2sin2dwhereis the eccentricity of the ellipse, where.54.Find the total length of the astroid,where,