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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Question
![Let \( f \) and \( g \) be continuous functions. If
\[
\int_{-6}^{-5} f(x) dx = 3, \quad \int_{-6}^{-5} g(x) dx = 5
\]
and \( D \) is the rectangle: \(-6 \leq x \leq -5, \, -6 \leq y \leq -5\),
then
\[
\iint_D f(x)g(y) \, dA =
\]
### Explanation of the Problem
This problem involves the evaluation of a double integral over a rectangular region \( D \). The functions \( f \) and \( g \) are given as continuous, and we are given the integral of each over a specified interval. Specifically, we are looking for the value of the integral:
\[
\iint_D f(x)g(y) \, dA
\]
### Evaluation Steps
1. **Integral of \( f(x) \):** We know that the integral of \( f(x) \) from \(-6\) to \(-5\) is 3:
\[
\int_{-6}^{-5} f(x) \, dx = 3
\]
2. **Integral of \( g(y) \):** We are also given that the integral of \( g(y) \) from \(-6\) to \(-5\) is 5:
\[
\int_{-6}^{-5} g(y) \, dy = 5
\]
3. **Double Integral \( \iint_D f(x)g(y) \, dA \):** To find the double integral over the region \( D \), we utilize the fact that:
\[
\iint_D f(x)g(y) \, dA = \left( \int_{-6}^{-5} f(x) \, dx \right) \times \left( \int_{-6}^{-5} g(y) \, dy \right)
\]
4. **Final Calculation:** Plug in the values:
\[
\iint_D f(x)g(y) \, dA = 3 \times 5 = 15
\]
Therefore, the value of the double integral is \( \boxed{15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F442a294d-0278-4659-8cf9-c6ac571e9e15%2F956c8213-5061-45ec-aeb1-3b864ab496d9%2Fkyf2tle_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f \) and \( g \) be continuous functions. If
\[
\int_{-6}^{-5} f(x) dx = 3, \quad \int_{-6}^{-5} g(x) dx = 5
\]
and \( D \) is the rectangle: \(-6 \leq x \leq -5, \, -6 \leq y \leq -5\),
then
\[
\iint_D f(x)g(y) \, dA =
\]
### Explanation of the Problem
This problem involves the evaluation of a double integral over a rectangular region \( D \). The functions \( f \) and \( g \) are given as continuous, and we are given the integral of each over a specified interval. Specifically, we are looking for the value of the integral:
\[
\iint_D f(x)g(y) \, dA
\]
### Evaluation Steps
1. **Integral of \( f(x) \):** We know that the integral of \( f(x) \) from \(-6\) to \(-5\) is 3:
\[
\int_{-6}^{-5} f(x) \, dx = 3
\]
2. **Integral of \( g(y) \):** We are also given that the integral of \( g(y) \) from \(-6\) to \(-5\) is 5:
\[
\int_{-6}^{-5} g(y) \, dy = 5
\]
3. **Double Integral \( \iint_D f(x)g(y) \, dA \):** To find the double integral over the region \( D \), we utilize the fact that:
\[
\iint_D f(x)g(y) \, dA = \left( \int_{-6}^{-5} f(x) \, dx \right) \times \left( \int_{-6}^{-5} g(y) \, dy \right)
\]
4. **Final Calculation:** Plug in the values:
\[
\iint_D f(x)g(y) \, dA = 3 \times 5 = 15
\]
Therefore, the value of the double integral is \( \boxed{15
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