Evaluate the surface integral ff 3x² ds, where S is part of the surface of a paraboloid with parameterization r(u, v) = (vcos (u), vsin (u), v²) where 0 ≤ u≤ and 0 ≤ v ≤ 1. Round your answer to two decimal places if necessary. Provide your answer below: ff 3x² ds units² ≈ [
Evaluate the surface integral ff 3x² ds, where S is part of the surface of a paraboloid with parameterization r(u, v) = (vcos (u), vsin (u), v²) where 0 ≤ u≤ and 0 ≤ v ≤ 1. Round your answer to two decimal places if necessary. Provide your answer below: ff 3x² ds units² ≈ [
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 Problem: Paraboloid Surface**
**Problem Statement:**
Evaluate the surface integral \(\iint_S 3x^2 \, dS\), where \(S\) is part of the surface of a paraboloid with parameterization:
\[
\mathbf{r}(u, v) = \langle v \cos(u), v \sin(u), v^2 \rangle
\]
where \(0 \leq u \leq \frac{\pi}{4}\) and \(0 \leq v \leq 1\).
Round your answer to two decimal places if necessary.
**Answer Box:**
\[
\iint_S 3x^2 \, dS \approx \, \boxed{ \ \ \ \ \ } \, \text{units}^2
\]
In this problem, the surface integral involves calculating \(\iint_S 3x^2 \, dS\) over a specified region on the surface of a paraboloid. The parameterization \(\mathbf{r}(u, v)\) uses parameters \(u\) and \(v\) to describe points on the paraboloid. The ranges of \(u\) and \(v\) are given, which define the portion of the paraboloid's surface to be integrated over.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13d2fa52-7679-4dbf-8ccf-e0669bc97f10%2F577bcb76-65f0-4dce-88b4-a8000a432fc0%2Fqk5pf8m_processed.png&w=3840&q=75)
Transcribed Image Text:**Surface Integral Problem: Paraboloid Surface**
**Problem Statement:**
Evaluate the surface integral \(\iint_S 3x^2 \, dS\), where \(S\) is part of the surface of a paraboloid with parameterization:
\[
\mathbf{r}(u, v) = \langle v \cos(u), v \sin(u), v^2 \rangle
\]
where \(0 \leq u \leq \frac{\pi}{4}\) and \(0 \leq v \leq 1\).
Round your answer to two decimal places if necessary.
**Answer Box:**
\[
\iint_S 3x^2 \, dS \approx \, \boxed{ \ \ \ \ \ } \, \text{units}^2
\]
In this problem, the surface integral involves calculating \(\iint_S 3x^2 \, dS\) over a specified region on the surface of a paraboloid. The parameterization \(\mathbf{r}(u, v)\) uses parameters \(u\) and \(v\) to describe points on the paraboloid. The ranges of \(u\) and \(v\) are given, which define the portion of the paraboloid's surface to be integrated over.
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