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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![**Surface Integral Evaluation Example**
**Problem Statement:**
Evaluate the surface integral over the surface \( S \):
\[
\iint_S f(x, y) \, dS
\]
where the function is given by:
\[
f(x, y) = xy
\]
**Surface Parameterization:**
The surface \( S \) is parameterized by:
\[
\mathbf{r}(u, v) = 4 \cos u \, \mathbf{i} + 4 \sin u \, \mathbf{j} + v \mathbf{k}
\]
**Parameter Domain:**
The parameters \( u \) and \( v \) are constrained by the following inequalities:
\[
0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq 1
\]
**Note:**
- This exercise involves calculating a surface integral, which is a type of integral used to find quantities (such as area or flux) over a curved surface in three-dimensional space.
- The parameterization \( \mathbf{r}(u, v) \) describes how points on the surface \( S \) are defined in terms of the parameters \( u \) and \( v \).
- The ranges for \( u \) and \( v \) specify the bounds of the region covered on the surface.
To solve, consider evaluating the integrand using the parameterization and compute the differential surface area element \( dS \). Then integrate over the given parameter domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa01b9ab7-eada-4439-963e-a6f55d227ef9%2F93ff6a02-0a14-4cdd-9313-29fb2eaf4589%2Fs6l1x9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Surface Integral Evaluation Example**
**Problem Statement:**
Evaluate the surface integral over the surface \( S \):
\[
\iint_S f(x, y) \, dS
\]
where the function is given by:
\[
f(x, y) = xy
\]
**Surface Parameterization:**
The surface \( S \) is parameterized by:
\[
\mathbf{r}(u, v) = 4 \cos u \, \mathbf{i} + 4 \sin u \, \mathbf{j} + v \mathbf{k}
\]
**Parameter Domain:**
The parameters \( u \) and \( v \) are constrained by the following inequalities:
\[
0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq 1
\]
**Note:**
- This exercise involves calculating a surface integral, which is a type of integral used to find quantities (such as area or flux) over a curved surface in three-dimensional space.
- The parameterization \( \mathbf{r}(u, v) \) describes how points on the surface \( S \) are defined in terms of the parameters \( u \) and \( v \).
- The ranges for \( u \) and \( v \) specify the bounds of the region covered on the surface.
To solve, consider evaluating the integrand using the parameterization and compute the differential surface area element \( dS \). Then integrate over the given parameter domain.
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