Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
Related questions
Question
![### Finding the Exact Length of a Curve
#### Problem Statement:
No calculators. Find the exact length of the curve given by:
\[ y = 2 + 2x^{3/2}, \quad 0 \leq x \leq 1 \]
#### Instructions:
To find the length of the curve described by the function \( y = 2 + 2x^{3/2} \) over the interval \([0, 1]\), you can use the arc length formula for a curve in Cartesian coordinates.
The arc length \( L \) of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]
1. **Compute the derivative** \( \frac{dy}{dx} \):
\[ y = 2 + 2x^{3/2} \]
\[ \frac{dy}{dx} = \frac{d}{dx} \left( 2 + 2x^{3/2} \right) \]
\[ \frac{dy}{dx} = 0 + 2 \cdot \frac{3}{2}x^{1/2} \]
\[ \frac{dy}{dx} = 3x^{1/2} \]
2. **Substitute** \( \frac{dy}{dx} \) into the arc length formula:
\[ L = \int_{0}^{1} \sqrt{1 + \left( 3x^{1/2} \right)^2 } \, dx \]
\[ L = \int_{0}^{1} \sqrt{1 + 9x} \, dx \]
3. **Evaluate the integral**:
To evaluate this integral, one way is to use a trigonometric substitution or a standard integration technique.
Let's proceed with the integration:
\[ L = \int_{0}^{1} \sqrt{1 + 9x} \, dx \]
For simplicity, one can perform this in detail using proper integration methods. However, the primary steps have been outlined to provide insight into the process.
This process will yield the exact length of the given curve from \( x = 0 \) to \( x = 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e042ae-8c1a-4cae-92ef-f152732755cf%2F68ab8aa2-7a05-4cf8-b7a4-122cea9c5fe2%2Fqkxtfbb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Exact Length of a Curve
#### Problem Statement:
No calculators. Find the exact length of the curve given by:
\[ y = 2 + 2x^{3/2}, \quad 0 \leq x \leq 1 \]
#### Instructions:
To find the length of the curve described by the function \( y = 2 + 2x^{3/2} \) over the interval \([0, 1]\), you can use the arc length formula for a curve in Cartesian coordinates.
The arc length \( L \) of a function \( y = f(x) \) from \( x = a \) to \( x = b \) is given by:
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]
1. **Compute the derivative** \( \frac{dy}{dx} \):
\[ y = 2 + 2x^{3/2} \]
\[ \frac{dy}{dx} = \frac{d}{dx} \left( 2 + 2x^{3/2} \right) \]
\[ \frac{dy}{dx} = 0 + 2 \cdot \frac{3}{2}x^{1/2} \]
\[ \frac{dy}{dx} = 3x^{1/2} \]
2. **Substitute** \( \frac{dy}{dx} \) into the arc length formula:
\[ L = \int_{0}^{1} \sqrt{1 + \left( 3x^{1/2} \right)^2 } \, dx \]
\[ L = \int_{0}^{1} \sqrt{1 + 9x} \, dx \]
3. **Evaluate the integral**:
To evaluate this integral, one way is to use a trigonometric substitution or a standard integration technique.
Let's proceed with the integration:
\[ L = \int_{0}^{1} \sqrt{1 + 9x} \, dx \]
For simplicity, one can perform this in detail using proper integration methods. However, the primary steps have been outlined to provide insight into the process.
This process will yield the exact length of the given curve from \( x = 0 \) to \( x = 1 \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL