text: # 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.
  
  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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explanation: The text extract contains a detailed math question about calculating the volume of a solid formed by rotating a shaded region about the x-axis. The region is defined by the intersection of a parabola \( y = x^2 - 3x \) and a line \( x + y - 3 = 0 \). The user is unsure about the correct approach when part of the area is under the x-axis and provides a potential solution of \( \frac{128}{5}\pi \). The extract also includes a detailed step-by-step solution to the problem, breaking it down into three integrals.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
  
  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.
  
  Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a detailed math question about calculating the volume of a solid formed by rotating a shaded region about the x-axis. The region is defined by the intersection of a parabola \( y = x^2 - 3x \) and a line \( x + y - 3 = 0 \). The user is unsure about the correct approach when part of the area is under the x-axis and provides a potential solution of \( \frac{128}{5}\pi \). The extract also includes a detailed step-by-step solution to the problem, breaking it down into three integrals.
  
  Label: question-and-answer