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x (t) = 3t – t3 y (t) = 3t2 %3D 0 st< 3

x (t) = 3t – t3 y (t) = 3t2 %3D 0 st< 3

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find the exact area of the surface obtained by rotating the given curve about the x-axis.

### Parametric Equations The image shows a set of parametric equations describing a curve: - \( x(t) = 3t - t^3 \) - \( y(t) = 3t^2 \) The parameter \( t \) ranges from 0 to 3: - \( 0 \leq t \leq 3 \) ### Explanation These equations describe a parametric curve in a two-dimensional plane. The parameter \( t \) is typically considered as a time variable, which gives the trajectory of a point as it moves along the curve from \( t = 0 \) to \( t = 3 \). - The equation for \( x(t) \) involves both a linear component \( 3t \) and a cubic component \( -t^3 \), indicating that the x-coordinate will increase initially and then turn due to the cubic term. - The equation for \( y(t) \) is quadratic \( 3t^2 \), implying that the y-coordinate always increases or stays the same since it is always positive in the given range of \( t \). This setup is often used to study motion and trajectories within physics and mathematics.
Transcribed Image Text:### Parametric Equations The image shows a set of parametric equations describing a curve: - \( x(t) = 3t - t^3 \) - \( y(t) = 3t^2 \) The parameter \( t \) ranges from 0 to 3: - \( 0 \leq t \leq 3 \) ### Explanation These equations describe a parametric curve in a two-dimensional plane. The parameter \( t \) is typically considered as a time variable, which gives the trajectory of a point as it moves along the curve from \( t = 0 \) to \( t = 3 \). - The equation for \( x(t) \) involves both a linear component \( 3t \) and a cubic component \( -t^3 \), indicating that the x-coordinate will increase initially and then turn due to the cubic term. - The equation for \( y(t) \) is quadratic \( 3t^2 \), implying that the y-coordinate always increases or stays the same since it is always positive in the given range of \( t \). This setup is often used to study motion and trajectories within physics and mathematics.
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