Question
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nsider the curve for . Find the exact arc length of the curve.

Ask by Floyd Weber. in the United States
Jan 25,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The exact arc length of the curve is units.

Solution

To find the exact arc length of the curve for , we’ll use the arc length formula for a function :

Step 1: Compute the First Derivative

Given:
Differentiate with respect to :

Step 2: Compute

Step 3: Set Up the Arc Length Integral

Thus, the arc length integral becomes:

Step 4: Evaluate the Integrals

Compute from 1 to 3:
Compute from 1 to 3:

Step 5: Combine the Results

Final Answer:
The exact arc length is 14⁄3 units.

Answered by UpStudy AI and reviewed by a Professional Tutor

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Simplify this solution

Extra Insights

To find the arc length of the curve from to , we use the arc length formula:
First, we need to calculate :
  1. Differentiate :
  2. Now, simplify :
  3. Next, compute :
  4. Substitute into the arc length formula:
  5. Simplify the term under the square root:
    Therefore,
  6. Now we can rewrite the arc length:
  7. Now, split the integral:
  8. Calculate :
  9. Calculate :
  10. Combine the results:
Thus, the exact arc length of the curve from to is

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