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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![On this educational webpage, we explore double integrals with an example expression.
The expression we are analyzing is a double integral defined as follows:
\[
\int_{1}^{2} \int_{-u^2 - 1}^{-u} (8uv) \, dv \, du
\]
This notation represents the double integral of the function \(8uv\) with respect to \(v\) and \(u\). Here, the integral is evaluated in two stages: the inner integral is with respect to \(v\) and the outer integral is with respect to \(u\).
- **Inner Integral:** \(\int_{-u^2 - 1}^{-u} (8uv) \, dv\)
- **Outer Integral:** \(\int_{1}^{2} \, du\)
The limits of integration for \(v\) are from \(-u^2 - 1\) to \(-u\) and for \(u\) are from 1 to 2.
### Detailed Explanation
1. **First Integrate with respect to \(v\):**
- Evaluate the inner integral \(\int_{-u^2 - 1}^{-u} (8uv) \, dv\) which requires finding the antiderivative of \(8uv\) with respect to \(v\).
2. **Then Integrate with respect to \(u\):**
- Once the inner integral is solved, substitute the result into the outer integral and integrate with respect to \(u\) over the interval from 1 to 2.
### Step-by-step Solution
1. **Find the Antiderivative with respect to \(v\):**
\[
\int (8uv) \, dv = 4uv^2
\]
- Because \(v^2/2\) is the antiderivative of \(v\).
2. **Evaluate the Inner Integral:**
- Substitute the limits for \(v\) into the antiderivative:
\[
\left[ 4uv^2 \right]_{-u^2-1}^{-u}
= 4u(-u)^2 - 4u(-u^2-1)^2
\]
3. **Simplify the Expression:**
- This involves algebraic simplification which should make the resulting function of \(u\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa90f48aa-fe1f-4446-8e54-ba82906f66bc%2Fe57fd4f0-7584-47d2-85f4-f32e53ec6250%2Fe7wrfx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this educational webpage, we explore double integrals with an example expression.
The expression we are analyzing is a double integral defined as follows:
\[
\int_{1}^{2} \int_{-u^2 - 1}^{-u} (8uv) \, dv \, du
\]
This notation represents the double integral of the function \(8uv\) with respect to \(v\) and \(u\). Here, the integral is evaluated in two stages: the inner integral is with respect to \(v\) and the outer integral is with respect to \(u\).
- **Inner Integral:** \(\int_{-u^2 - 1}^{-u} (8uv) \, dv\)
- **Outer Integral:** \(\int_{1}^{2} \, du\)
The limits of integration for \(v\) are from \(-u^2 - 1\) to \(-u\) and for \(u\) are from 1 to 2.
### Detailed Explanation
1. **First Integrate with respect to \(v\):**
- Evaluate the inner integral \(\int_{-u^2 - 1}^{-u} (8uv) \, dv\) which requires finding the antiderivative of \(8uv\) with respect to \(v\).
2. **Then Integrate with respect to \(u\):**
- Once the inner integral is solved, substitute the result into the outer integral and integrate with respect to \(u\) over the interval from 1 to 2.
### Step-by-step Solution
1. **Find the Antiderivative with respect to \(v\):**
\[
\int (8uv) \, dv = 4uv^2
\]
- Because \(v^2/2\) is the antiderivative of \(v\).
2. **Evaluate the Inner Integral:**
- Substitute the limits for \(v\) into the antiderivative:
\[
\left[ 4uv^2 \right]_{-u^2-1}^{-u}
= 4u(-u)^2 - 4u(-u^2-1)^2
\]
3. **Simplify the Expression:**
- This involves algebraic simplification which should make the resulting function of \(u\
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