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COs t 3 sin t (a) wh at kind of cur ve is this ? (6) Sketch itt graph. (C) (c) Find the area enclosed. (d) Use symmetry and Simpson's Rule (n= 2) to estimate the circum ference.

COs t 3 sin t (a) wh at kind of cur ve is this ? (6) Sketch itt graph. (C) (c) Find the area enclosed. (d) Use symmetry and Simpson's Rule (n= 2) to estimate the circum ference.

Calculus: Early Transcendentals
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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Can anyone help please me with these three problems? I am stuck! ( These aren’t graded questions)
### Parametric Equations and Their Analysis Consider the parametric equations: \[ x = 2 \cos t, \quad y = 3 \sin t, \quad 0 \leq t \leq 2\pi \] 1. **(a) What kind of curve is this?** 2. **(b) Sketch its graph.** 3. **(c) Find the area enclosed.** 4. **(d) Use symmetry and Simpson's Rule (n=2) to estimate the circumference.** --- **Explanation:** - The equations describe a parametric curve where \( x \) and \( y \) vary with \( t \). Given the trigonometric functions involved, this is likely an ellipse oriented along the standard axes. - To sketch the graph, plot points for values of \( t \) ranging from 0 to \( 2\pi \). The values of \( x \) and \( y \) will trace out the ellipse. - Calculating the enclosed area involves integrating the function or using known formulas for areas of ellipses. In this case, the area \( A \) of an ellipse is given by \( A = \pi \times a \times b \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. - For the circumference, Simpson's Rule, a numerical integration method, can provide an estimate by taking advantage of the symmetry of the ellipse. This exercise might include diagrams or graphs illustrating the shape and characteristics of the ellipse as derived from these equations.
Transcribed Image Text:### Parametric Equations and Their Analysis Consider the parametric equations: \[ x = 2 \cos t, \quad y = 3 \sin t, \quad 0 \leq t \leq 2\pi \] 1. **(a) What kind of curve is this?** 2. **(b) Sketch its graph.** 3. **(c) Find the area enclosed.** 4. **(d) Use symmetry and Simpson's Rule (n=2) to estimate the circumference.** --- **Explanation:** - The equations describe a parametric curve where \( x \) and \( y \) vary with \( t \). Given the trigonometric functions involved, this is likely an ellipse oriented along the standard axes. - To sketch the graph, plot points for values of \( t \) ranging from 0 to \( 2\pi \). The values of \( x \) and \( y \) will trace out the ellipse. - Calculating the enclosed area involves integrating the function or using known formulas for areas of ellipses. In this case, the area \( A \) of an ellipse is given by \( A = \pi \times a \times b \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. - For the circumference, Simpson's Rule, a numerical integration method, can provide an estimate by taking advantage of the symmetry of the ellipse. This exercise might include diagrams or graphs illustrating the shape and characteristics of the ellipse as derived from these equations.
### Exercise 2 **Equation:** \( r = 3 \), \( 0 \leq \theta \leq 2\pi \) **Tasks:** (a) Sketch the graph in the \( r\theta \)-plane. (b) Sketch the graph in the \( xy \)-plane. (c) Find the circumference. --- ### Exercise 3 **Equation:** \( r = 2 + \cos \theta \), \( 0 \leq \theta \leq 2\pi \) **Tasks:** (a) Sketch the graph in the \( r\theta \)-plane. (b) Sketch the graph in the \( xy \)-plane. (c) Find the area enclosed.
Transcribed Image Text:### Exercise 2 **Equation:** \( r = 3 \), \( 0 \leq \theta \leq 2\pi \) **Tasks:** (a) Sketch the graph in the \( r\theta \)-plane. (b) Sketch the graph in the \( xy \)-plane. (c) Find the circumference. --- ### Exercise 3 **Equation:** \( r = 2 + \cos \theta \), \( 0 \leq \theta \leq 2\pi \) **Tasks:** (a) Sketch the graph in the \( r\theta \)-plane. (b) Sketch the graph in the \( xy \)-plane. (c) Find the area enclosed.
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