Volume of Solid of Revolution about Oblique Axis - Kyle's Question

In summary, the volume of the solid formed by revolving the region bounded by the graphs of y=x and y=x^2 about the line y=x is pi/(30sqrt(2)). This can be found using the formula V=\frac{\pi}{\left(m^2+1 \right)^{\frac{3}{2}}}\int_{x_i}^{x_f} \left(f(x)-mx-b \right)^2\left(1+mf'(x) \right)\,dx, where m=1, b=0, and f(x)=x^2. The final result is found to be pi/(30sqrt(2)) after solving the integral.
  • #1
MarkFL
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Here is the question:

Find the volume of the solid formed by revolving about y=x?

Find the volume of the solid formed by revolving the region bounded by the graphs of y=x and y=x^2 about the line y=x

Additional Details:

The book's given answer is pi/(30sqrt(2))

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello Kyle,

Please see http://mathhelpboards.com/math-notes-49/solid-revolution-about-oblique-axis-rotation-6683.html for the development of the general formula:

\(\displaystyle V=\frac{\pi}{\left(m^2+1 \right)^{\frac{3}{2}}}\int_{x_i}^{x_f} \left(f(x)-mx-b \right)^2\left(1+mf'(x) \right)\,dx\)

We first find $x_i$ and $x_f$:

\(\displaystyle x^2=x\)

\(\displaystyle x(x-1)=0\)

Hence:

$x_i=0,\,x_f=1$

We are given:

$m=1,\,b=0,\,f(x)=x^2\implies f'(x)=2x$

Hence:

\(\displaystyle V=\frac{\pi}{2\sqrt{2}}\int_{0}^{1} \left(x^2-x \right)^2(1+2x)\,dx\)

Expanding the integrand, we have:

\(\displaystyle V=\frac{\pi}{2\sqrt{2}}\int_{0}^{1} 2x^5-3x^4+x^2 \,dx\)

Applying the FTOC, we find:

\(\displaystyle V=\frac{\pi}{2\sqrt{2}}\left[\frac{1}{3}x^6-\frac{3}{5}x^5+\frac{1}{3}x^3 \right]_{0}^{1}\)

\(\displaystyle V=\frac{\pi}{2\sqrt{2}}\left(\frac{1}{3}-\frac{3}{5}+\frac{1}{3} \right)=\frac{\pi}{2\sqrt{2}}\left(\frac{1}{15} \right)=\frac{\pi}{30\sqrt{2}}\)
 

FAQ: Volume of Solid of Revolution about Oblique Axis - Kyle's Question

1) What is the "Volume of Solid of Revolution about Oblique Axis"?

The Volume of Solid of Revolution about Oblique Axis is a mathematical concept that involves calculating the volume of a 3-dimensional solid formed by rotating a 2-dimensional shape about an oblique axis, which is not perpendicular to the base of the shape.

2) How is the volume of a solid of revolution about an oblique axis calculated?

The volume is calculated by using the method of cylindrical shells, which involves integrating the cross-sectional area of the shape with respect to the axis of revolution. This results in a volume formula that takes into account the oblique axis and the shape's original dimensions.

3) Who is Kyle and what is his question about the volume of solid of revolution about an oblique axis?

Kyle is an individual who may have posed a specific question about this mathematical concept. Without further context, it is not possible to determine the specific question that Kyle may have asked.

4) What are some real-world applications of the volume of solid of revolution about oblique axis?

This mathematical concept has various applications in fields such as engineering, architecture, and physics. For example, it can be used to calculate the volume of a curved structure such as an arch, or to determine the volume of a container with an oblique axis.

5) Are there any limitations or assumptions when using the volume of solid of revolution about oblique axis?

Like any mathematical concept, there are certain limitations and assumptions that should be considered when using the volume of solid of revolution about oblique axis. These may include assuming the shape has a continuous cross-section, and the axis of revolution is a straight line. It is also important to ensure that the correct limits of integration are used when calculating the volume.

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