MHB Find the Volume of Rotated Shaded Region Bounded by y=x^2+1, y=5, and y-axis

[lilukelay]

In the diagram, the shaded region is bounded by the parabola y = x2 + 1, the y-axis and the line y = 5.
Find the volume of the solid formed when the shaded region is rotated about the y-axis.
Got no diagram but limits will be 2-0 coz its on right side

 

[MarkFL]

Using the first quadrant area, and the disk method, we may state:

$\displaystyle dV=\pi x^2\,dy$

Since we have $\displaystyle x^2=y-1$ we may state:

$\displaystyle dV=\pi(y-1)\,dy$

And by integration, we have:

$\displaystyle V=\pi\int_1^5 y-1\,dy$

Using the shell method, we find:

$\displaystyle dV=2\pi x(5-y)\,dx$

Since $\displaystyle y=x^2+1$, we may state:

$\displaystyle dV=2\pi x(5-(x^2+1))\,dx=2\pi(4x-x^3)\,dx$

And by integration, we have:

$\displaystyle V=2\pi\int_0^2 4x-x^3\,dx$