graphing calculator is recommended. Consider the following. x = t² - 7t, y = t5³, 1st≤4 Write an integral expression that represents the length of the curve described by the parametric equations. 19 dt Ise technology to find the length of the curve. (Round your answer to four decimal places.)
graphing calculator is recommended. Consider the following. x = t² - 7t, y = t5³, 1st≤4 Write an integral expression that represents the length of the curve described by the parametric equations. 19 dt Ise technology to find the length of the curve. (Round your answer to four decimal places.)
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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![### Parametric Curve Length Calculation
A graphing calculator is recommended for this exercise.
Consider the following parametric equations:
\[ x = t^2 - 7t, \]
\[ y = t^5, \]
\[ 1 \leq t \leq 4 \]
#### Write an integral expression that represents the length of the curve described by these parametric equations.
To find the length of the curve, we utilize the integral formula for the length of a curve given by parametric equations:
\[ L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt \]
For the given parametric equations:
\[ \frac{dx}{dt} = \frac{d}{dt}(t^2 - 7t) = 2t - 7 \]
\[ \frac{dy}{dt} = \frac{d}{dt}(t^5) = 5t^4 \]
Substituting these into the integral formula:
\[ L = \int_{1}^{4} \sqrt{(2t - 7)^2 + (5t^4)^2} \, dt \]
#### Use technology to find the length of the curve. (Round your answer to four decimal places.)
\[ \boxed{\phantom{0}} \]
Please use a graphing calculator or appropriate computational tools to evaluate the integral and find the length of the curve. Be sure to round your answer to four decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06f55c03-0ff2-4f1b-a27d-6e1bf7c58dcc%2F9516c7a0-bc29-4bfb-9e22-f853747f7559%2F856g7rj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Parametric Curve Length Calculation
A graphing calculator is recommended for this exercise.
Consider the following parametric equations:
\[ x = t^2 - 7t, \]
\[ y = t^5, \]
\[ 1 \leq t \leq 4 \]
#### Write an integral expression that represents the length of the curve described by these parametric equations.
To find the length of the curve, we utilize the integral formula for the length of a curve given by parametric equations:
\[ L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt \]
For the given parametric equations:
\[ \frac{dx}{dt} = \frac{d}{dt}(t^2 - 7t) = 2t - 7 \]
\[ \frac{dy}{dt} = \frac{d}{dt}(t^5) = 5t^4 \]
Substituting these into the integral formula:
\[ L = \int_{1}^{4} \sqrt{(2t - 7)^2 + (5t^4)^2} \, dt \]
#### Use technology to find the length of the curve. (Round your answer to four decimal places.)
\[ \boxed{\phantom{0}} \]
Please use a graphing calculator or appropriate computational tools to evaluate the integral and find the length of the curve. Be sure to round your answer to four decimal places.
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