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 Statement:**
Find the double integral \( \int_{1}^{2} \int_{0}^{1} (3x^2 + 4y^3) \, dy \, dx \).
**Solution:**
The expression requires evaluating a double integral of the function \( 3x^2 + 4y^3 \) over the rectangle defined by \( 1 \leq x \leq 2 \) and \( 0 \leq y \leq 1 \).
**Steps:**
1. **Inner Integral:**
Evaluate the inner integral with respect to \( y \):
\[
\int_{0}^{1} (3x^2 + 4y^3) \, dy
\]
2. **Outer Integral:**
After finding the result of the inner integral, evaluate the outer integral over \( x \):
\[
\int_{1}^{2} \ldots \, dx
\]
The final result will give the value of the double integral.
This process is a fundamental application of the Fubini's Theorem, which allows us to compute such integrals by iteratively integrating along each dimension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F358deae0-b7d2-45f9-8056-32e7a5350251%2F52495d33-c809-4c79-893e-c7f5ac03a8d1%2F6h3fz4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the double integral \( \int_{1}^{2} \int_{0}^{1} (3x^2 + 4y^3) \, dy \, dx \).
**Solution:**
The expression requires evaluating a double integral of the function \( 3x^2 + 4y^3 \) over the rectangle defined by \( 1 \leq x \leq 2 \) and \( 0 \leq y \leq 1 \).
**Steps:**
1. **Inner Integral:**
Evaluate the inner integral with respect to \( y \):
\[
\int_{0}^{1} (3x^2 + 4y^3) \, dy
\]
2. **Outer Integral:**
After finding the result of the inner integral, evaluate the outer integral over \( x \):
\[
\int_{1}^{2} \ldots \, dx
\]
The final result will give the value of the double integral.
This process is a fundamental application of the Fubini's Theorem, which allows us to compute such integrals by iteratively integrating along each dimension.
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