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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 12.1.86
Chapter 12, Problem 12.1.86

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

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Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\): \[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Find the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) by differentiating each component with respect to \(t\). Use the product rule where necessary since both \(x\) and \(y\) are products of functions of \(t\).
Compute \(\left(\frac{dx}{dt}\right)^2\) and \(\left(\frac{dy}{dt}\right)^2\) and then add them together inside the square root to form the integrand: \[\sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\]
Simplify the expression under the square root as much as possible to make the integral easier to evaluate.
Set up the definite integral for the arc length from \(t=0\) to \(t=\pi\): \[L = \int_0^{\pi} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\] This integral can then be evaluated using appropriate techniques or numerical methods.

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with parametric forms is essential for calculating properties like arc length.
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Parameterizing Equations

Arc Length Formula for Parametric Curves

The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula measures the distance along the curve by summing infinitesimal line segments, requiring differentiation of both x and y with respect to t.
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Arc Length of Parametric Curves

Differentiation of Parametric Functions

To apply the arc length formula, one must compute the derivatives dx/dt and dy/dt accurately. This involves using standard differentiation rules, including the product and chain rules, especially when x(t) and y(t) are products or compositions of functions. Correct differentiation is crucial for evaluating the integral correctly.
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Related Practice
Textbook Question

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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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

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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

15–22. Sets in polar coordinates Sketch the following sets of points.


1 < r < 2 and π/6 ≤ θ ≤ π/3

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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