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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how do i solve the attached calculus question?
![**Problem:**
Calculate the integral of \( f(x, y, z) = 9x^2 + 9y^2 + z^8 \) over the curve \( \mathbf{c}(t) = (\cos t, \sin t, t) \) for \( 0 \leq t \leq \pi \).
\[
\int_{c} (9x^2 + 9y^2 + z^8) \, ds = \, \text{{[Input box for the answer]}}
\]
**Explanation:**
The integral to be calculated is a line integral over the specified curve \( \mathbf{c}(t) \). The function \( f(x, y, z) \) is defined in terms of \( x^2 \), \( y^2 \), and \( z^8 \), where each variable \( x, y, \) and \( z \) is parametrized by, respectively, \(\cos t\), \(\sin t\), and \(t\).
This problem involves evaluating the line integral of a scalar field over a given parametric curve. The curve is expressed in terms of trigonometric and linear functions of \( t \). The limits of integration are from \( t = 0 \) to \( t = \pi \).
For such a problem, one would substitute the parametric equations into the function and compute the integral, considering the differential arc length \( ds \) along the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2F3b5909f5-2f5f-4ee7-a798-e3f59fe89e7e%2Fjv5rfmr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Calculate the integral of \( f(x, y, z) = 9x^2 + 9y^2 + z^8 \) over the curve \( \mathbf{c}(t) = (\cos t, \sin t, t) \) for \( 0 \leq t \leq \pi \).
\[
\int_{c} (9x^2 + 9y^2 + z^8) \, ds = \, \text{{[Input box for the answer]}}
\]
**Explanation:**
The integral to be calculated is a line integral over the specified curve \( \mathbf{c}(t) \). The function \( f(x, y, z) \) is defined in terms of \( x^2 \), \( y^2 \), and \( z^8 \), where each variable \( x, y, \) and \( z \) is parametrized by, respectively, \(\cos t\), \(\sin t\), and \(t\).
This problem involves evaluating the line integral of a scalar field over a given parametric curve. The curve is expressed in terms of trigonometric and linear functions of \( t \). The limits of integration are from \( t = 0 \) to \( t = \pi \).
For such a problem, one would substitute the parametric equations into the function and compute the integral, considering the differential arc length \( ds \) along the curve.
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