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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Find the lengths of the curves
![### Integral Representation and Domain Explanation
#### Equation:
\[ y = \int_{-2}^{x} \sqrt{3t^4 - 1} \, dt, \quad -2 \leq x \leq -1 \]
#### Explanation:
The equation presented above is an integral function, where the variable \( y \) is defined as the definite integral of the function \( \sqrt{3t^4 - 1} \) with respect to \( t \), evaluated from the lower limit \(-2\) to the upper limit \( x \).
#### Integral Function Components:
- **Integral Symbol (\( \int \))**: Indicates that the function \( y \) is an integral.
- **Limits of Integration (\(-2\) to \( x \))**: These specify the range over which the integration is performed. In this case, the integration starts at \( t = -2 \) and ends at \( t = x \).
- **Integrand (\( \sqrt{3t^4 - 1} \))**: This is the function inside the integral that is being integrated with respect to \( t \). Specifically, it is the square root of the polynomial \( 3t^4 - 1 \).
- **Differential (\( dt \))**: Indicates that the integration is performed with respect to the variable \( t \).
#### Domain Definition:
The domain for \( x \) in this function is given by \(-2 \leq x \leq -1\), meaning that \( x \) can have any value within the closed interval from \(-2\) to \(-1\).
### Detailed Graphical Analysis (Not Provided):
While the provided equation is not accompanied by a graph, consider creating a detailed graph plotting the function \( y = \int_{-2}^{x} \sqrt{3t^4 - 1} dt \) over the closed interval \([-2, -1]\) for better visualization:
1. **Compute Specific Values**: Calculate the integral for specific values of \( x \) within the interval \([-2, -1]\).
2. **Plotting**: Graphically represent the computed \( y \) values against the corresponding \( x \) values.
3. **Behavior Analysis**: Investigate how the integral behaves as \( x \) moves from \(-2\) to \(-1\).
This](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a617c73-16c8-405c-b9c4-72dd4741d892%2Fe65ed5fc-c1c3-4ce3-a557-2cb7e8970475%2Fsrqx55d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Integral Representation and Domain Explanation
#### Equation:
\[ y = \int_{-2}^{x} \sqrt{3t^4 - 1} \, dt, \quad -2 \leq x \leq -1 \]
#### Explanation:
The equation presented above is an integral function, where the variable \( y \) is defined as the definite integral of the function \( \sqrt{3t^4 - 1} \) with respect to \( t \), evaluated from the lower limit \(-2\) to the upper limit \( x \).
#### Integral Function Components:
- **Integral Symbol (\( \int \))**: Indicates that the function \( y \) is an integral.
- **Limits of Integration (\(-2\) to \( x \))**: These specify the range over which the integration is performed. In this case, the integration starts at \( t = -2 \) and ends at \( t = x \).
- **Integrand (\( \sqrt{3t^4 - 1} \))**: This is the function inside the integral that is being integrated with respect to \( t \). Specifically, it is the square root of the polynomial \( 3t^4 - 1 \).
- **Differential (\( dt \))**: Indicates that the integration is performed with respect to the variable \( t \).
#### Domain Definition:
The domain for \( x \) in this function is given by \(-2 \leq x \leq -1\), meaning that \( x \) can have any value within the closed interval from \(-2\) to \(-1\).
### Detailed Graphical Analysis (Not Provided):
While the provided equation is not accompanied by a graph, consider creating a detailed graph plotting the function \( y = \int_{-2}^{x} \sqrt{3t^4 - 1} dt \) over the closed interval \([-2, -1]\) for better visualization:
1. **Compute Specific Values**: Calculate the integral for specific values of \( x \) within the interval \([-2, -1]\).
2. **Plotting**: Graphically represent the computed \( y \) values against the corresponding \( x \) values.
3. **Behavior Analysis**: Investigate how the integral behaves as \( x \) moves from \(-2\) to \(-1\).
This
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