Surface Area of an Ellipse Obtained by Rotation

In summary, the conversation discusses finding the surface area of a surface obtained by rotating an ellipse around the x-axis. The formula for calculating the surface area is given, but the speaker is unsure how to evaluate it. The conversation also includes a suggestion to use a substitution to simplify the integral.
  • #1
renyikouniao
41
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A surface is obtained by rotating around the x-axis the arc over the integral(-1,0.5) of an ellipse given by:
x^2+4y^2=1

What is its surface area?

Here's my solution:

I use the equation:
S=integral( upper bound: a lower bound: b ) 2(pi)y*[1+(f'(x))^2]^0.5 dx

Since x^2+4y^2=1
y=[(1-x^2)/4]^0.5
dy/dx=-x/[2(1-x^2)^0.5]

S=integral (upper bound: 0.5 lower bound: -1) 2(pi)[(1-x^2)/4]^0.5 * [1 -x/[2(1-x^2)^0.5]]^0.5

And I have no idea how to evaluate this whole thing...Am I right so far?
 
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  • #2
Re: Surface area

I find (using the top-half of the ellipse):

\(\displaystyle f(x)=\frac{1}{2}\sqrt{1-x^2}\)

\(\displaystyle f'(x)=-\frac{x}{2\sqrt{1-x^2}}\)

and so the surface of rotation is given by:

\(\displaystyle S=2\pi\int_{-1}^{\frac{1}{2}} \frac{1}{2}\sqrt{1-x^2}\sqrt{1+\frac{x^2}{4\left(1-x^2 \right)}}\,dx\)

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

At this point, I suggest the substitution:

\(\displaystyle x=\frac{2}{\sqrt{3}}\sin(\theta)\)

Can you proceed?
 

FAQ: Surface Area of an Ellipse Obtained by Rotation

What is the formula for finding the surface area of an ellipse obtained by rotation?

The formula for finding the surface area of an ellipse obtained by rotation is A = πab, where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

How is the surface area of an ellipse obtained by rotation different from that of a regular ellipse?

The surface area of an ellipse obtained by rotation is larger than that of a regular ellipse because it takes into account the additional curvature created by rotating the ellipse around its minor axis.

Can the surface area of an ellipse obtained by rotation be negative?

No, the surface area of an ellipse obtained by rotation cannot be negative as it is a measure of the total area enclosed by the curved surface and must be a positive value.

How does changing the angle of rotation affect the surface area of an ellipse obtained by rotation?

Changing the angle of rotation will affect the surface area of an ellipse obtained by rotation by altering the curvature and elongation of the ellipse, resulting in a larger or smaller surface area.

Is there a real-world application for the surface area of an ellipse obtained by rotation?

Yes, the surface area of an ellipse obtained by rotation has practical applications in engineering and architecture, such as calculating the surface area of a curved dome or the volume of a storage tank with a curved bottom.

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