Find the volume of the solid obtained by rotating the region in the 1st quadrant bounded by y=x^6, y=1, and the y-axis about the line y=−2.?

By Washer Method,
V = ∫ π ( R^2 - r^2 ) dx
R = outer radius = 1 - (-2) = 3
r = inner radius = x^6 - (-2) = x^6 + 2

V
= ∫ π ( R^2 - r^2 ) dx , from x = 0 to x = 1
= π * ∫ [ 3^2 - ( x^6 + 2 )^2 ] dx
= π * ∫ ( - x^12 - 4x^6 + 5 ) dx
= π * [ - (1/13)x^13 - (4/7)x^7 + 5x ] , from x = 0 to x = 1
= π * ( - 1/13 - 4/7 + 5 )
= 396π / 91 ..... Ans