Calculate surface area by rotating the curve

Question

You make a new hat by rotating the curve f(x)=1/64X^7 +1/32X^5+3/8X^3-3/4X^2-2X+2   -1<= x <=2 around the line x=-1. Calculate the surface area of new hat.

Answer 1

For a curve y=f(x) rotated around a vertical line x=0 , the surface area is

where

•r(x) is the horizontal distance from the point (x,f(x)) to the line x=0 .

Here, a=-1, so

r(x)= ⌊ x- (-1) ⌊=x+⌈1

(because on ⌈-1, 2⌉, we have x+1≥0).

s =2 π ∫ (x+1) √1+(7/64 x² + 5/34 x⁴ + 9/8 x² -3/2 x -2 ) ² dx on ⌈-1, 2⌉