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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.52
Chapter 12, Problem 12.1.52

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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Recall the general parametric equations for a circle centered at \((h, k)\) with radius \(r\): \[x = h + r \cos(t), \quad y = k + r \sin(t)\] where \(t\) is the parameter, usually representing the angle in radians.
Identify the given values from the problem: the center is \((-2, 2)\) and the radius is \(6\). Substitute these into the general form: \[x = -2 + 6 \cos(t), \quad y = 2 + 6 \sin(t)\]
Since the problem asks for the lower half of the circle, determine the range of \(t\) that corresponds to the lower semicircle. The lower half means \(y \leq 2\), so \(\sin(t) \leq 0\). This occurs when \(t\) is between \(\pi\) and \(2\pi\).
Confirm the orientation is counterclockwise. The parameter \(t\) increasing from \(\pi\) to \(2\pi\) moves the point along the circle in the counterclockwise direction on the lower half.
Write the final parametric equations with the parameter interval: \[x = -2 + 6 \cos(t), \quad y = 2 + 6 \sin(t), \quad \pi \leq t \leq 2\pi\]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations of a Circle

A circle centered at (h, k) with radius r can be represented parametrically as x = h + r cos(t) and y = k + r sin(t), where t is the parameter. This form allows describing the position of points on the circle as t varies, typically over an interval of length 2π.
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Parameterizing Equations of Circles & Ellipses

Parameter Interval and Orientation

The parameter interval defines which portion of the curve is traced and in what direction. For counterclockwise orientation, t usually increases from 0 to 2π. To represent only the lower half of the circle, t is restricted to the interval where y-values correspond to the lower semicircle.
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Eliminating the Parameter

Adjusting Parametric Equations for Specific Curve Segments

To describe a specific segment of a curve, such as the lower half of a circle, the parameter range is limited accordingly. Understanding how the sine and cosine functions behave over intervals helps select the correct parameter bounds to capture the desired portion of the curve.
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Parameterizing Equations
Related Practice
Textbook Question

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

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Textbook Question

81–88. Arc length Find the arc length of the following curves on the given interval.


x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π

Textbook Question

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 1 + sin θ and r = 1 + cos θ

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Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=t,y= √(4−t²) a

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b