Is there someway to find the exact area of a blob using integrals?

In summary, in order to calculate the area of a random blob or pie slice, you can use relative integrals by writing the boundary of the shape in terms of integrable functions. For a pie slice, you can integrate in polar coordinates with boundaries of radius (0,r_o) and angle (0, theta). However, this can be challenging and the key is to express the boundary in terms of integrable functions.
  • #1
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Someone told me Isaac Newton developed some infinitesimal triangle series to find the area of a random blob, but I think there might be some way to do it this way by drawing many lines from a central point to the edge, although that would make more of a pie slice, but is there some way to calculate the area of that pie slice using relative integrals?
 
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  • #2
Yes, it is simple if you can write the boundary of the "blob" in terms of integrable functions! That's the hard part.
 
  • #3
HallsofIvy said:
Yes, it is simple if you can write the boundary of the "blob" in terms of integrable functions! That's the hard part.

But what about it being in the shape of a pie slice? The range is just from x1 to x1?
 
  • #4
A pie slice would be simple. You integrate in polar coordinates over the radius and angle. The boundaries are then (assuming a normal slice of pie): radius (0,r_o) and angle (0, theta)
 

1. Can integrals be used to find the exact area of a blob?

Yes, integrals can be used to find the exact area of a blob if the boundaries of the blob can be expressed as a mathematical function.

2. What is the process for finding the area of a blob using integrals?

The process involves setting up a definite integral with the boundaries of the blob as the limits of integration and the function representing the blob as the integrand. The integral is then solved using integration techniques.

3. Are there any limitations to using integrals to find the area of a blob?

Yes, the limitations include the need for a mathematical function to represent the boundaries of the blob and the complexity of the function, which can make the integral difficult or impossible to solve analytically.

4. How accurate is using integrals to find the area of a blob?

Using integrals to find the area of a blob can provide an exact and accurate result if the integral is solved correctly. However, any errors in the setup or solving of the integral can lead to inaccurate results.

5. Can integrals be used to find the area of a blob in three-dimensional space?

Yes, integrals can be extended to three-dimensional space to find the volume of a blob. However, the process becomes more complex as it involves setting up a triple integral with three variables representing the three dimensions.

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