Answer:
44.18 cubic unit
Step-by-step explanation:
The intersection of the curve [tex]y = x^2[/tex] and [tex]y = 8-x^2[/tex] is
[tex]x^2 = 8 - x^2[/tex]
[tex]2x^2 = 8[/tex]
[tex]x^2 = 4[/tex]
[tex]x = \pm 2[/tex]
Since we are rotating the region where [tex]x \geq 0.5[/tex] we will use the intersection x = 2 and y = 4
Using the shell method with x ranges from 0.5 to 2. The volume of the shell would be
[tex]V = \int\limits^2_{0.5} {2\pi x h} \, dx[/tex]
where h is the difference between the y-coordinates of the curves, in other words
[tex]h = 8 - x^2 - x^2 = 8 - 2x^2[/tex]
Plug h into the V integral and we have
[tex]V = \int\limits^2_{0.5} {2\pi x (8 - 2x^2)} \, dx\\V = 4\pi\int\limits^2_{0.5} {(4x - x^3)} \, dx\\V = 4\pi[2x^2 - x^4/4]^2_{0.5}\\V = 4\pi(2*2^2 - 2^4/4 - 2*0.5^2 + 0.5^4/4)\\V = 4\pi(8 - 4 - 0.5 + 0.015625) \approx 44.18[/tex]
