Find the length of the curve for the interval 1 ≤ x ≤ 9 3 = √ √ √ ² ³ - 1 d dt 1 Select the correct answer. 1 484 OL= OL= OL= 5 O4 = 484 13 484 5 484 y
Find the length of the curve for the interval 1 ≤ x ≤ 9 3 = √ √ √ ² ³ - 1 d dt 1 Select the correct answer. 1 484 OL= OL= OL= 5 O4 = 484 13 484 5 484 y
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding the Length of a Curve
In this problem, you are required to find the length of the curve for the given interval \(1 \leq x \leq 9\). The curve is defined by the equation:
\[ y = \int_{1}^{x} \sqrt{t^3 - 1} \, dt \]
You need to find the arc length \( L \) for this curve over the specified interval.
#### Select the correct answer from the options below:
- \( \circ \quad L = \frac{1}{84} \)
- \( \circ \quad L = \frac{84}{13} \)
- \( \circ \quad L = \frac{84}{5} \)
- \( \circ \quad L = \frac{5}{84} \)
### Explanation of the Formula for Arc Length
To find the length of a curve given by \( y = \int_{a}^{x} f(t) \, dt \) from \( x = a \) to \( x = b \):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Here,
\[ \frac{dy}{dx} = \sqrt{x^3 - 1} \]
Therefore,
\[ L = \int_{1}^{9} \sqrt{1 + (\sqrt{x^3 - 1})^2} \, dx \]
\[ L = \int_{1}^{9} \sqrt{1 + (x^3 - 1)} \, dx \]
\[ L = \int_{1}^{9} \sqrt{x^3} \, dx \]
\[ L = \int_{1}^{9} x^{3/2} \, dx \]
After evaluating the integral, you can match the result with the correct answer from the options provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2Fdefad368-df85-40e2-9fd9-1c16a0553054%2Fgkjriwf_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Length of a Curve
In this problem, you are required to find the length of the curve for the given interval \(1 \leq x \leq 9\). The curve is defined by the equation:
\[ y = \int_{1}^{x} \sqrt{t^3 - 1} \, dt \]
You need to find the arc length \( L \) for this curve over the specified interval.
#### Select the correct answer from the options below:
- \( \circ \quad L = \frac{1}{84} \)
- \( \circ \quad L = \frac{84}{13} \)
- \( \circ \quad L = \frac{84}{5} \)
- \( \circ \quad L = \frac{5}{84} \)
### Explanation of the Formula for Arc Length
To find the length of a curve given by \( y = \int_{a}^{x} f(t) \, dt \) from \( x = a \) to \( x = b \):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Here,
\[ \frac{dy}{dx} = \sqrt{x^3 - 1} \]
Therefore,
\[ L = \int_{1}^{9} \sqrt{1 + (\sqrt{x^3 - 1})^2} \, dx \]
\[ L = \int_{1}^{9} \sqrt{1 + (x^3 - 1)} \, dx \]
\[ L = \int_{1}^{9} \sqrt{x^3} \, dx \]
\[ L = \int_{1}^{9} x^{3/2} \, dx \]
After evaluating the integral, you can match the result with the correct answer from the options provided.
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