Area of cylinder sliced by sphere

In summary, the task is to calculate the area of a cylinder defined by the equation $$x^{2}+y^{x}=ax$$ and sliced by a sphere with the equation $$x^{2}+y^{2}+z^{2}=a^{2}$$. The graph provided shows the intersection of the two shapes. The speaker suggests using double integrals to solve a similar problem with a sphere sliced by a cylinder, and mentions using cylindrical coordinates and translating the shapes to simplify the calculation. They believe the area of the cylinder sliced by a sphere can be found in a similar manner.
  • #1

etf

179
2
Hi!
Here is my task:
Calculate area of cylinder $$x^{2}+y^{x}=ax$$ sliced by sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$.
Here is graph:

111.jpg


How to do it? If problem was "Calculate area of sphere $$x^{2}+y^{2}+z^{2}=a^{2}$$ sliced by cylinder $$x^{2}+y^{x}=ax$$" I would solve it using double integrals ($$P=\iint\sqrt{1+(\frac{\partial z}{\partial x})^{2}+(\frac{\partial z}{\partial y})^{2}}dxdy$$ over region $$x^{2}+y^{x}=ax$$). That area would be this (I think):

222.jpg


123.jpg


But what would be area of cylinder sliced by sphere? Maybe this:

444.jpg
 
Last edited:
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  • #2
First correct the exponent of y in your equation of the cylinder, it is 2. Yes, it is the surface area of the cylinder contained within the sphere. Then, at first glance, I would translate the whole thing left by a/2 so that the axis of the cylinder passes through the origin and use cylindrical coordinates. r is constant, the limits on your polar angle are a full circle, and the limits on z (as a function of angle) are found from the intersection of the two equations. Should be straightforward (I didn't actually do it).
 

1. What is the formula for finding the area of a cylinder sliced by a sphere?

The formula for finding the area of a cylinder sliced by a sphere is 2πrh, where r is the radius of the cylinder and h is the height of the cylinder.

2. Can the area of a cylinder sliced by a sphere be greater than the area of the cylinder itself?

Yes, the area of a cylinder sliced by a sphere can be greater than the area of the cylinder itself. This is because the sphere creates a curved surface on the cylinder, increasing its surface area.

3. How does the radius of the sphere affect the area of the cylinder sliced by it?

The radius of the sphere directly affects the area of the cylinder sliced by it. As the radius of the sphere increases, the area of the cylinder also increases.

4. Are there any real-world applications of calculating the area of a cylinder sliced by a sphere?

Yes, there are many real-world applications of calculating the area of a cylinder sliced by a sphere. For example, this calculation is used in engineering and architecture when designing structures such as water tanks or silos.

5. Can the area of a cylinder sliced by a sphere be negative?

No, the area of a cylinder sliced by a sphere cannot be negative. This is because area is a measure of the amount of space occupied by a shape, and it cannot have a negative value.

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