How do you find the area of the surface generated by rotating the curve about the y-axis #x=t+1, y=1/2t^2+t, 0<=t<=2#? Calculus Applications of Definite Integrals Determining the Surface Area of a Solid of Revolution 1 Answer Ultrilliam Jun 25, 2018 #= (2 pi)/3( 10 sqrt(10) - 2 sqrt(2)) " sq units"# Explanation: #bbr(t) = << t+1, 1/2 t(t+2) >>, qquad 0<=t<=2# #ds = sqrt(dot x^2 + dot y^2) \ dt# #dS = ds * 2 pi x = 2 pi (t+1) sqrt((1)^2 + (t+1)^2) \ dt# #= 2 pi (t+1) sqrt(t^2 + 2t + 2) \ dt# #S =2 pi int_0^2 dt qquad (t+1) sqrt(t^2 + 2t + 2) # # =2 pi int_0^2 dt qquad d/dt ( 1/3(t^2 + 2t + 2)^(3/2)) # # =(2 pi)/3 [ (t^2 + 2t + 2)^(3/2)]_0^2 # # =(2 pi)/3 ( 10^(3/2) - 2^(3/2))# #= (2 pi)/3( 10 sqrt(10) - 2 sqrt(2)) approx 60.3 " sq units"# Answer link Related questions How do you find the surface area of a solid of revolution? How do you find the surface area of the solid obtained by rotating about the #y#-axis the region... How do you find the surface area of the solid obtained by rotating about the #x#-axis the region... How do you find the surface area of the solid obtained by rotating about the #x#-axis the region... How do you find the surface area of the solid obtained by rotating about the #y#-axis the region... How do you find the surface area of the solid obtained by rotating about the #x#-axis the region... How do you find the surface area of the solid obtained by rotating about the #x#-axis the region... How do you find the surface area of the part of the circular paraboloid #z=x^2+y^2# that lies... How do you determine the surface area of a solid revolved about the x-axis? How do you find the centroid of the quarter circle of radius 1 with center at the origin lying... See all questions in Determining the Surface Area of a Solid of Revolution Impact of this question 558 views around the world You can reuse this answer Creative Commons License