Computing Arc Length and Cannot Solve the Integral

In summary, the conversation discussed finding the length of a polar curve, specifically the cardioid r=1+cosθ. The relevant equation for calculating the length was provided, and the attempt at a solution involved manipulating the given functions. However, the integrals could not be solved due to a lack of familiarity with trigonometric identities.
  • #1
Voodoo583
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Homework Statement



Find the length of the polar curve.

Complete the cardioid r=1+cosθ

Homework Equations



L=∫αβ√[f(θ)2+f'(θ)2] dθ

The Attempt at a Solution



Given f(θ)2 is equal to cos2θ+2cosθ+1

and f'(θ)2=sin2θ

I arrive at the integral ∫αβ√[2cosθ+2] dθ which I cannot for the life of me solve for. It's been a while since my calc II class so I flipped through my textbook to see if I could find a solution. I cannot do u substitution as far as I can tell nor integration by parts and I couldn't find anything similar to the form above in the table of integrals.
 
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  • #2

1. What is computing arc length?

Computing arc length is a mathematical process used to find the length of a curve on a graph. This is commonly done by using the integral function to calculate the arc length.

2. How is arc length calculated using integrals?

To compute arc length using integrals, the curve is divided into small segments and the length of each segment is approximated using the Pythagorean theorem. The lengths of these segments are then added together using the integral function to get a more accurate measurement of the arc length.

3. Why is it sometimes impossible to solve the integral for computing arc length?

The integral function is used to calculate the area under a curve, but it is not always possible to find an exact solution for the integral. In some cases, the integral may involve complex functions or involve an infinite number of terms, making it impossible to solve.

4. What are some alternative methods for finding arc length?

If the integral cannot be solved, other methods such as numerical integration or using a computer program may be used to approximate the arc length. Additionally, in some cases, it may be possible to find the arc length using geometric formulas or by using trigonometric functions.

5. How accurate are the results when computing arc length using integrals?

The accuracy of the results depends on the precision of the integral function and the number of segments used to approximate the arc length. Generally, the more segments used, the more accurate the result will be. However, there may still be some degree of error due to the approximation process.

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