What is the surface area produced by rotating #f(x)=x^3-x^2+1, x in [0,3]# around the x-axis?

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Answer 1 · 2017-02-15T16:35:36

#9837/70pi=441.5# cubic units, nearly.

graph{(x^3-x^2+1-y)(x-3+.001y)y(x-.001y)=0 [0, 4, -1, 20]}

Volume =#pi int y^2 dx#, with #y = x^3-x^2+1 #and x from 0 to 3

#=pi int( x^3-x^2+1)^2 dx#, with x from 0 t0 3

#=pi int (x^6+x^4+1-2x^5-2x^2+2x^3) dx#, for the limits

#=pi[x^7/7+x^5/5+x-1/3x^6-2/3x^3+1/2x^4]#, between x = 0 and 3

#=pi[(2187/7+243/5-243-18+81/2)-(0)]#

#=9837/70pi#

#(140.529)(3.1416)=441.5 cubic units, nearly.