Breaking up the semi-circle into infinitesimally small rings

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In summary, the speaker attempted to solve a problem involving a semi-circle by breaking it up into small rings and using a formula for integration. However, the resulting answer did not match what was expected. They then asked for assistance and mentioned the possibility of starting incorrectly.
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spencerw105
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I attempted to solve this problem by breaking up the semi-circle into infinitesimally small rings of mass dm and width dr a distance r away from the center [0<r<R]. I then wrote dm in terms of the area: dm=pi*r*dr. Then, I plugged into the formula di=integral(dm*r^2) and integrated from 0 to R giving me .25*pi*R^4. This is not what I would have gotten for a complete disk though I would have gotten .5*pi*R^4 using the same process. Can someone please help I don't even know if I started correctly.
 
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spencerw105 said:
I plugged into the formula di=integral(dm*r^2) and integrated from 0 to R giving me .25*pi*R^4. This is not what I would have gotten for a complete disk though I would have gotten .5*pi*R^4 using the same process. Can someone please help I don't even know if I started correctly.
You have left out a constant factor for density. What happens when you include that and express it in terms of the mass and area of the object?
 

Related to Breaking up the semi-circle into infinitesimally small rings

1. What does it mean to "break up the semi-circle into infinitesimally small rings"?

Breaking up the semi-circle into infinitesimally small rings means dividing the semi-circle into an infinite number of smaller and smaller sections, each with a slightly different radius. This concept is often used in calculus and geometry to calculate the area or circumference of complex shapes.

2. Why is breaking up the semi-circle into infinitesimally small rings useful?

This technique allows us to approximate the area or circumference of a semi-circle with greater accuracy. By dividing the shape into smaller and smaller sections, we can get closer to the exact value and minimize any errors in our calculations.

3. How do you actually break up a semi-circle into infinitesimally small rings?

The process involves taking the semi-circle and dividing it into an infinite number of concentric circles, each with a slightly different radius. This creates a series of rings that get closer and closer to the original semi-circle. In mathematical terms, this is known as taking the limit as the width of each ring approaches zero.

4. Can you give an example of using this concept in real-world applications?

Sure, one example is calculating the volume of a sphere. By breaking up the sphere into infinitesimally small rings and using calculus to integrate them, we can find the volume with greater precision than using traditional geometry methods.

5. Are there any limitations to using this method?

While breaking up the semi-circle into infinitesimally small rings can provide more accurate calculations, it requires a lot of mathematical knowledge and can be time-consuming. Additionally, in some cases, the limit of the width of each ring may not approach the exact value and could result in errors. It is important to understand the limitations and assumptions when using this technique.

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