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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![4. (Section 16.3) Evaluate the double integral
\[
\int\int_{R} e^{-3x^2 - 3y^2} \, dA;
\]
where \( R = \{ (x, y) \, | \, x^2 + y^2 \leq 4, \, x \leq 0, \, y \geq 0 \} \).
**Explanation:**
This is a double integral problem where you need to evaluate the integral of the function \( e^{-3x^2 - 3y^2} \) over the region \( R \). The region \( R \) is defined by two conditions:
1. \( x^2 + y^2 \leq 4 \), which describes a circle of radius 2 centered at the origin.
2. \( x \leq 0 \) and \( y \geq 0 \), which restricts the region to the second quadrant, including the part of the circle that lies in this quadrant.
The detailed solution involves setting up the double integral with these limits and evaluating it over the defined region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F251f9165-68ff-4948-bb45-e4b672f91d1b%2F71ea9f84-d55e-46e4-9078-d0310bc10f4d%2Fnqvfmd5_processed.png&w=3840&q=75)
Transcribed Image Text:4. (Section 16.3) Evaluate the double integral
\[
\int\int_{R} e^{-3x^2 - 3y^2} \, dA;
\]
where \( R = \{ (x, y) \, | \, x^2 + y^2 \leq 4, \, x \leq 0, \, y \geq 0 \} \).
**Explanation:**
This is a double integral problem where you need to evaluate the integral of the function \( e^{-3x^2 - 3y^2} \) over the region \( R \). The region \( R \) is defined by two conditions:
1. \( x^2 + y^2 \leq 4 \), which describes a circle of radius 2 centered at the origin.
2. \( x \leq 0 \) and \( y \geq 0 \), which restricts the region to the second quadrant, including the part of the circle that lies in this quadrant.
The detailed solution involves setting up the double integral with these limits and evaluating it over the defined region.
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