Answer:
Volume of solid is [tex]\frac{40}{3}\pi[/tex]
Step-by-step explanation:
We need to find the volume of solid formed by rotating the region bounded by the graph of [tex]y=1+sqrt{x}[/tex], y-axis and the line y=3 about the line y=5.
Please see the attachment for figure.
Using Shell method, [tex]V=\int_a^b\pi (R^2-r^2)dx[/tex]
where,
a=0 (Lower limit of solid)
b=4 (Upper limit of solid)
[tex]R=4-\sqrt{x}[/tex] (Outer Radius of Shell)
r=2 (Inner radius of shell)
dx is thickness of shell
Volume of shell, dV=Area of shell x Thickness
Volume of solid [tex]V=\int dV[/tex]
[tex]V=\int_0^4 \pi [(4-\sqrt{x})^2-2^2]dx[/tex]
[tex]V=\int_0^4 \pi (16+x-8\sqrt{x}-4)dx[/tex]
[tex]V=\pi (12x+\frac{x^2}{2}-\frac{16}{3}x^{3/2}|_0^4[/tex]
[tex]V=\pi (48+8-\frac{128}{3})[/tex]
[tex]V=\frac{40}{3}\pi[/tex]
Thus, Volume of solid is [tex]\frac{40}{3}\pi[/tex]
