find the volume of the solid formed by rotating the region enclosed by x=0, x=1, y=0, y=9+x^7 about the y axis

Question

find the volume of the solid formed by rotating the region enclosed by x=0, x=1, y=0, y=9+x^7 about the y axis

Answer 1

Use shell method to evaluate this volume. The shell method essentially calculates the volume as a sum of volumes of infinitesimally thin shells. Imagine the infinitesimal rectangle from x to x+dx rotated around y-axis. The volume of this shell is given by:

 

dV=2πx*h(x)dx, where h(x) is the height of the shell, in your case it is x⁷+9. Total volume is:

 

V=∫₀¹ 2πx*(x⁷+9)dx=2π(x⁹/9+9x²/2)|₀¹=2π(1/9+9/2)=83π/9