# Volume of rotation about x-axis with part of the area under the x-axis

part of the graph y=x^2-3x is shown below. The line x+y-3=0 is drawn through the point (-1,4) such that this line intersects the parabola again at the point (3,0) as shown below.

The shaded region in the diagram above is rotated about the x-axis. Calculate the exact volume of the solid formed.

Please help me out with this question. Usually, I know how to do volume of rotation about the x-axis. however, this question has part of the area under the x-axis. Doing it normally with (volume above - volume below) gives me 128/5(pi), but I am unsure if this is correct, when part of the area is under the axis.

## 1 Answer

First find $x \in (-1,3)$ such that

\[ |x^2-3x|=|3-x|.\]

Then

\[ |x^2-3x|=3x-x^2=3-x=|3-x| \Rightarrow x^2-4x+3=0.\]

\[ \Rightarrow (x-1)(x-3)=0 \Rightarrow x=1, x=3.\]

So $x=1$ is the only solution in $(-1,3)$. Note that for $x\in (1,3)$ we have $|x^2-3x|>|3-x|$ and for $x\in (0,1)$ we have $|x^2-3x|<|3-x|$. When rotating about $x$-axis, the volume will be created by the curve that has the larger distance from the $x-$axis, i.e. by $y=x^2-3x$ on $(1,3)$ and by $y=3-x$ on $(0,1)$. So we need to break up this into three integrals:

\[\text{Volume}=\int_0^1 \pi (3-x)^2dx+\int_1^3 \pi (3x-x^2)^2dx+\int_{-1}^{0}\pi [(3-x)^2-(x^2-3x)^2]dx.\]

Note that on $(0,1)$ and $(1,3)$ we should use the disk method, but on $(-1,0)$ we use the washer method.

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