Question

the region in the first quadrant enclosed by the system of equations is rotated about the x-axis. the volume of the solid generated is: y=0, x=5, and y= 1/(sqrt(1+x))

Answer 1

With the disk method, the volume is given by the integral


`\displaystyle\pi\int_(x=0)^(x=5)\left(\frac1{√(1+x)}\right)^2\,\mathrm dx=\pi\int_0^5(\mathrm dx)/(1+x)`

`=\displaystyle\pi\int_(t=1)^(t=6)\frac{\mathrm dt}t`

where the substitution
`t=x+1` was made, giving
`\mathrm dt=\mathrm dx`.


`=\displaystyle\pi\ln|t|\bigg|_(t=1)^(t=6)`

`=\pi(\ln6-\ln1)`

`=\pi\ln6`