Struggling to determine the bounds for this volume by integration problem

I have $$y=|x|+1, 2x+3y=6\text{ about }x=-20.$$

I require to set up an indispensable (yet not review it) that will certainly offer the quantity of the strong when the area confined by those contours is rotated concerning the offered line. I need to make use of either the covering or washing machine is method.

I have actually been attempting to make use of the covering method (I assume) yet I am having problem establishing the indispensable. Just how do I identify the bounds for it? I'm not exactly sure just how I get the numbers for "top" and also "bottom" either. I addressed for y in the $2x+3y=6$ formula and also obtained $y=-(2/3)x+2$ yet just how do I recognize where that belongs in the indispensable?

Thanks for any kind of aid

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2022-06-07 14:40:03
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Hint: If you outline both contours, they create a triangular. This is the location that you are rotating, so find the works with of the 3 edges of the triangular. If you make use of round coverings, the minimum $y$ value is the start of your assimilation and also the maximum $y$ is completion. After that for an offered $y$ because array, you require to find the minimum $x$ and also maximum $x$. The minimum will constantly be along $y=1-x$ (why?) yet the maximum will certainly change from one line to the various other component means along, so you possibly intend to divide the indispensable there.
Whatever the recommended approaches, the most basic means to locate this quantity $V$ is to make use of Guldin is regulation: $V$ amounts to the location $A$ of the offered triangular $\Delta$ times the area of the circle defined by the center of gravity $S$ of $\Delta$. The vertices of $\Delta$ being $(0,1)$, $({3\over5}, {8\over5})$ and also $(-3,4)$ we get $A={9\over5}$ and also $S=(-{4\over5},{11\over5})$. This indicates $$V= {9\over5} \cdot 2\pi (-{4\over5}+20) ={1728\over25}\ \pi\ .$$