1. ## Volume by rotation

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A triangle is formed by the lines x = 1; x + y = 7; and
3x-4y=7. Rotate the triangle about the line x=-1 to create a volume. Find volume by setting up definite integral.

Thanks out there to anyone who can help.

2. Hey, how about the washer method? Can you use the figure below and the following to do it? Pretty sure that's it. Did it kinda' quick. Double-check it ok.

$\displaystyle V=\pi\int_{-14}^{1}\left[ (f+1)^2-(g+1)^2\right] dy+\pi\int_1^6 \left[(f+1)^2-4\right] dy$

$\displaystyle f$ and $\displaystyle g$ are the equations for the two lines in terms of $\displaystyle y$ right? For example $\displaystyle f(y)=(7-y)$.

3. Thanks, that looks right.

4. after looking at it further, I figured that the +1 in the figure is the extra length on the radius b/c of rotation around the line x=-1. Did I figure right? Also, where did the negative four come from? Does it come from the inner cylinder formed with a volume of 4? Thanks again.

5. Originally Posted by michaelaparker
after looking at it further, I figured that the +1 in the figure is the extra length on the radius b/c of rotation around the line x=-1. Did I figure right? Also, where did the negative four come from? Does it come from the inner cylinder formed with a volume of 4? Thanks again.
The extra 1 comes from the rotation about a line 1 unit from the y-axis. Also, the 4 comes from the radius squared of the red line in the plot below revolved around the line x=-1.