Huzafa's question at Yahoo Answers regarding a solid of revolution

In summary, we can find the volume of the solid obtained by revolving the plane area between the graph y=1-x^2 and the x-axis around the y-axis using either the disk method or the shell method. Both methods yield a volume of π/2.
  • #1
MarkFL
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Here is the question:

Find volume of solid obtained by revolving around y-axis the plane area btw the graph y=1-x^2 and the x-axis?

I have posted a link there to this topic so the OP can see my work.
 
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  • #2
Hello huzafa,

The first thing I would do is draw a diagram of the region to be revolved. We need only concern ourselves with either the quadrant I area or the quadrant II area because we are revolving an even function about the $y$-axis. I will choose to plot the quadrant I area:

View attachment 974

Using the disk method, we observe that the volume of an arbitrary disk is:

\(\displaystyle dV=\pi r^2\,dy\)

where:

\(\displaystyle r=x\,\therefore\,r^2=x^2=1-y\)

and so we have:

\(\displaystyle dV=\pi(1-y)\,dy\)

Summing the disks by integration, we have:

\(\displaystyle V=\pi\int_0^1 1-y\,dy=\pi\int_0^1 u\,du=\frac{\pi}{2}\left[u^2 \right]_0^1=\frac{\pi}{2}\)

Using the shell method, we observe that the volume of an arbitrary shell is:

\(\displaystyle dV=2\pi rh\,dx\)

where:

\(\displaystyle r=x\)

\(\displaystyle h=y=1-x^2\)

and so we have:

\(\displaystyle dV=2\pi\left(x-x^3 \right)\,dx\)

Summing the shells by integration, we find:

\(\displaystyle V=2\pi\int_0^2 x-x^3\,dx=\frac{\pi}{2}\left[2x^2-x^4 \right]_0^1=\frac{\pi}{2}\)
 

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FAQ: Huzafa's question at Yahoo Answers regarding a solid of revolution

1. What is a solid of revolution?

A solid of revolution is a three-dimensional figure created by rotating a two-dimensional shape around an axis. The resulting shape is symmetrical and has a circular cross-section.

2. How do you find the volume of a solid of revolution?

To find the volume of a solid of revolution, you can use the formula V = π∫ba (f(x))2 dx, where a and b are the limits of integration and f(x) is the function that represents the cross-sectional area of the shape at a specific point along the axis of rotation.

3. Can any two-dimensional shape be rotated to create a solid of revolution?

Yes, any two-dimensional shape can be rotated to create a solid of revolution as long as the shape is continuous and has a closed perimeter.

4. What is the difference between a solid of revolution and a cylinder?

A cylinder is a specific type of solid of revolution where the base shape is a circle. However, a solid of revolution can have any two-dimensional base shape, such as a triangle, rectangle, or even a more complex curve.

5. How is a solid of revolution used in real life?

A solid of revolution has many practical applications, such as in engineering and architecture, where it is used to create objects with symmetrical shapes, such as wheels, pipes, and columns. It is also used in calculus to solve problems related to volume and surface area.

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