How to find the volume of the solid with integral?

Question

I need to find the volume of the solid obtained by rotating the y-axis: $$(y-1)^2 = x, x = 1$$

What definite integral should I use? With what boundaries? I tried this $\int_0^1(y-1)^4dy$,but I think it is wrong. I know that there is formula: $$2\pi\int_{y1}^{y2} y(x)*x dx$$ But how can I "pull out" $y(x)$ Help please.

Answer 1

Let's try with this one (cylinder method):

$$2\pi\int_0^1 x(2\sqrt x) dx$$

The correct formula is:

$$2\pi\int_{x_1}^{x_2} |f_1(x)-f_2(x)|\cdot x dx$$

In this case:

$$(y-1)^2 = x \implies y-1=\pm \sqrt x \implies y=\pm\sqrt x+1$$

$$f_1(x)=\sqrt x +1$$

$$f_2(x)=-\sqrt x +1$$