Question
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.
The exact arc length is 14⁄3 units.
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Extra Insights
To find the arc length of the curve
from
to
, we use the arc length formula:
First, we need to calculate
:
-
Differentiate
: -
Now, simplify
: -
Next, compute
: -
Substitute
into the arc length formula: -
Simplify the term under the square root:Therefore,
-
Now we can rewrite the arc length:
-
Now, split the integral:
-
Calculate
: -
Calculate
: -
Combine the results:
Thus, the exact arc length of the curve from
to
is