Solution #1.#
Using the washer type disk
#dV=pi(R^2-r^2)dh#
#V=int_0^1 pi((sqrt(y)+8)^2-(y^2+8)^2)dy#
#V=pi int_0^1 [y+16y^(1/2)+64-(y^4+16y^2+64)]dy#
#V=pi int_0^1 [y+16y^(1/2)-y^4-16y^2]dy#
#V=pi [y^2/2+32/3y^(3/2)-y^5/5-16/3y^3]_0^1#
#V=pi [(1)^2/2+32/3(1)^(3/2)-(1)^5/5-16/3(1)^3-0]#
#V=pi(1/2+32/3-1/5-16/3)#
#V=pi((15+320-6-160)/30)#
#color(red)(V=(169pi)/30)" "#cubic units
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Solution #2.#
Using cylindrical shell
#dV=2pi*r*h*dr#
#V=2pi int_0^1 (x+8)(sqrtx-x^2)*dx#
#V=2pi int_0^1 (x^(3/2)-x^3+8x^(1/2)-8x^2)dx#
#V=2pi [2/5x^(5/2)-x^4/4+16/3x^(3/2)-(8x^3)/3]_0^1#
#V=2pi [2/5(1)^(5/2)-(1)^4/4+16/3(1)^(3/2)-(8(1)^3)/3-0]#
#V=2pi [2/5-1/4+16/3-8/3]#
#V=2pi [(24-15+320-160)/60]#
#V=2pi [169/60]#
#color(red)(V=(169pi)/30)" "#cubic units
God bless....I hope the explanation is useful.
