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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![**Problem b**
Evaluate the double integral:
\[
\int\int_{T} xy^3 \, dA
\]
where \( T \) is the triangle with vertices \((-1, 0)\), \((0, 5)\), \((1, 0)\).
This integral represents the computation of a double integral over a triangular region \( T \) defined by its vertices in the plane. The function \( xy^3 \) is integrated with respect to area over this triangle.
### Explanation of the Triangle
- The vertex \((-1, 0)\) is located on the x-axis to the left of the origin.
- The vertex \( (0, 5) \) is located directly above the origin on the y-axis.
- The vertex \( (1, 0) \) is on the x-axis to the right of the origin.
This forms a right triangle with the line segment connecting these points. The base of the triangle runs horizontally along the x-axis, and the height reaches up to \( y = 5 \).
To solve this problem, you need to set up the integral with the correct limits of integration for \( x \) and \( y \), based on the geometry of the triangular region \( T \). This typically involves:
1. Determining the equations of the lines connecting the vertices.
2. Using these equations to find the bounds for \( y \) in terms of \( x \) or vice versa.
3. Integrating the function \( xy^3 \) over this region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29ea5d5b-5dae-48f3-a188-d0a58b2c7f90%2F6a9437e3-62cc-414c-a2bb-c913731b18d4%2F33vq32b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem b**
Evaluate the double integral:
\[
\int\int_{T} xy^3 \, dA
\]
where \( T \) is the triangle with vertices \((-1, 0)\), \((0, 5)\), \((1, 0)\).
This integral represents the computation of a double integral over a triangular region \( T \) defined by its vertices in the plane. The function \( xy^3 \) is integrated with respect to area over this triangle.
### Explanation of the Triangle
- The vertex \((-1, 0)\) is located on the x-axis to the left of the origin.
- The vertex \( (0, 5) \) is located directly above the origin on the y-axis.
- The vertex \( (1, 0) \) is on the x-axis to the right of the origin.
This forms a right triangle with the line segment connecting these points. The base of the triangle runs horizontally along the x-axis, and the height reaches up to \( y = 5 \).
To solve this problem, you need to set up the integral with the correct limits of integration for \( x \) and \( y \), based on the geometry of the triangular region \( T \). This typically involves:
1. Determining the equations of the lines connecting the vertices.
2. Using these equations to find the bounds for \( y \) in terms of \( x \) or vice versa.
3. Integrating the function \( xy^3 \) over this region.
![**Problem Statement:**
5. Compute the following double integrals:
a)
\[
\int \int_{R} ye^{xy} \, dA
\]
where \( R \) is the square \(-1 \leq x \leq 1\), \( 0 \leq y \leq 2 \).
**Explanation:**
This problem involves finding the double integral of the function \(ye^{xy}\) over the region \(R\), which is defined as the square \(-1 \leq x \leq 1\) and \(0 \leq y \leq 2\). The integration should be performed over the specified rectangular domain in the xy-plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29ea5d5b-5dae-48f3-a188-d0a58b2c7f90%2F6a9437e3-62cc-414c-a2bb-c913731b18d4%2Fosbl790c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
5. Compute the following double integrals:
a)
\[
\int \int_{R} ye^{xy} \, dA
\]
where \( R \) is the square \(-1 \leq x \leq 1\), \( 0 \leq y \leq 2 \).
**Explanation:**
This problem involves finding the double integral of the function \(ye^{xy}\) over the region \(R\), which is defined as the square \(-1 \leq x \leq 1\) and \(0 \leq y \leq 2\). The integration should be performed over the specified rectangular domain in the xy-plane.
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