Set up, but do not evaluate, an integral that represents the length of the parametric curve Select the correct answer. 10 10 5 10 10 + 32x (In 3)² dx 3 O√1 5 + 3* In 3 dx + 32x( *(In 3)² dx 3²* (In 3)² dx +10²* (In 10)² dx y=3*, 5 ≤ x ≤ 10.
Set up, but do not evaluate, an integral that represents the length of the parametric curve Select the correct answer. 10 10 5 10 10 + 32x (In 3)² dx 3 O√1 5 + 3* In 3 dx + 32x( *(In 3)² dx 3²* (In 3)² dx +10²* (In 10)² dx y=3*, 5 ≤ x ≤ 10.
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...
Related questions
Question
![### Calculating the Length of a Parametric Curve
#### Problem Statement:
Set up, but do not evaluate, an integral that represents the length of the parametric curve defined by:
\[ y = 3^x, \quad 5 \leq x \leq 10 \]
#### Select the correct answer:
1. \[
\int_{5}^{10} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
2. \[
\int_{5}^{10} \sqrt{1 + 3^x \ln 3} \, dx
\]
3. \[
\int_{10}^{5} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
4. \[
\int_{10}^{5} \sqrt{3^{2x} (\ln 3)^2} \, dx
\]
5. \[
\int_{5}^{3} \sqrt{1 + 10^{2x} (\ln 10)^2} \, dx
\]
#### Explanation:
To find the arc length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the following integral:
\[ \text{Arc length} = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Given \( y = 3^x \):
1. Compute \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = 3^x \ln 3
\]
2. Substitute into the arc length formula:
\[
\text{Arc length} = \int_{5}^{10} \sqrt{1 + \left(3^x \ln 3\right)^2} \, dx
\]
Thus, the correct integral is:
\[
\int_{5}^{10} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
So, the correct answer is the first option.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e8bfde-46ac-4815-81bc-d8be1d9c4f35%2F9af4b0bc-069f-415a-a16a-017b53cd8a10%2Fccp07id_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Length of a Parametric Curve
#### Problem Statement:
Set up, but do not evaluate, an integral that represents the length of the parametric curve defined by:
\[ y = 3^x, \quad 5 \leq x \leq 10 \]
#### Select the correct answer:
1. \[
\int_{5}^{10} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
2. \[
\int_{5}^{10} \sqrt{1 + 3^x \ln 3} \, dx
\]
3. \[
\int_{10}^{5} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
4. \[
\int_{10}^{5} \sqrt{3^{2x} (\ln 3)^2} \, dx
\]
5. \[
\int_{5}^{3} \sqrt{1 + 10^{2x} (\ln 10)^2} \, dx
\]
#### Explanation:
To find the arc length of a curve given by \( y = f(x) \) from \( x = a \) to \( x = b \), we use the following integral:
\[ \text{Arc length} = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
Given \( y = 3^x \):
1. Compute \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = 3^x \ln 3
\]
2. Substitute into the arc length formula:
\[
\text{Arc length} = \int_{5}^{10} \sqrt{1 + \left(3^x \ln 3\right)^2} \, dx
\]
Thus, the correct integral is:
\[
\int_{5}^{10} \sqrt{1 + 3^{2x} (\ln 3)^2} \, dx
\]
So, the correct answer is the first option.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning