I have a triangle on the $xy$ plane whos base's center is $x_1$ away from the $y$-axis horizontally. I want to rotate it around the y-axis and find the volume of the triangular doughnut shape.
My idea is to find the area of the triangle using $\frac{1}{2}bh$ and then multiply this area by the circumference of the circle with center at the y-axis and radius $x_1$.
So the triangular doughnut volume would be $\frac{1}{2}bh \cdot 2\pi r=bh \cdot \pi r$
Is this correct, if not is there a better method?
Your approach doesn't work: the outer parts of the triangle are being swept through more distance than the inner parts. Some integration is needed.
First, let's give some things names: Let $x_0$ be the closest your triangle gets to the $y$-axis (you've implicitly assumed that $x_0 > 0$, so I'll do the same), let $x_2$ be the furthest. Then let the vertices of your triangle be $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Let $f(t) = t(x_0,y_0) + (1-t)(x_1,y_1)$, $g(t) = t(x_1,y_1) + (1-t)(x_2,y_2)$, and $h(t) = t(x_0,y_0) + (1-t)(x_2,y_2)$ be paramaterisations of the three sides of the triangle.
Now, consider some cylinder around the $y$ axis of radius $r \in [x_0,x_2]$. The length of the line segment formed by intersecting $r$ with your triangle is given by:
$$l(r) = \left\{\array{\left|\dfrac{(x_1-r)y_0+ (r-x_0)y_1}{x_1-x_0} - \dfrac{(x_2-r)y_0+(r-x_0)y_2}{x_2-x_0}\right| &r < x_1\\\left|\dfrac{(x_2-r)y_1+ (r-x_1)y_2}{x_2-x_1} - \dfrac{(x_2-r)y_0+(r-x_0)y_2}{x_2-x_0}\right| & r \geq x_1}\right.$$
Thus, the surface area of the cylinder of revolution of that line segment is just $2\pi r$ times that. Our total surface is then the sum of all of these, so is given by
$$\int_{x_0}^{x_1}2\pi r\left|\dfrac{(x_1-r)y_0+ (r-x_0)y_1}{x_1-x_0} - \dfrac{(x_2-r)y_0+(r-x_0)y_2}{x_2-x_0}\right|\mathrm{d}r + \int_{\!\!x_1}^{\!\!x_2}2\pi r\left|\dfrac{(x_2-r)y_1+ (r-x_1)y_2}{x_2-x_1} - \dfrac{(x_2-r)y_0+(r-x_0)y_2}{x_2-x_0}\right|\mathrm{d}r.$$
(Which, if you bother to do all the integration, turns out to be equivalent to what you'd get out of Pappus' Second Theorem).