Textbook Question
Find the area of the region bounded by the astroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.1.84
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
Verified step by step guidance1
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 \(\frac{dy}{dt}\) for the given functions:
\[x = e^t \sin t, \quad y = e^t \cos t\]
Use the product rule for differentiation since both \(x\) and \(y\) are products of \(e^t\) and trigonometric functions.
Compute \(\frac{dx}{dt}\):
\[\frac{dx}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t\]
Compute \(\frac{dy}{dt}\):
\[\frac{dy}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t\]
Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length formula under the square root:
\[\sqrt{(e^t \sin t + e^t \cos t)^2 + (e^t \cos t - e^t \sin t)^2}\]
Simplify the expression inside the square root by expanding and combining like terms.
Set up the integral for the arc length over the interval \(0 \leq t \leq 2\pi\):
\[L = \int_0^{2\pi} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt\]
After simplification, evaluate or express the integral in a form ready for calculation.
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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 a curve as functions of a parameter, often denoted 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.
Recommended video:
Guided course
08:02
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 sums the infinitesimal distances along the curve, providing the total length over the interval.
Recommended video:
06:29Arc Length of Parametric Curves
Differentiation of Exponential and Trigonometric Functions
Calculating dx/dt and dy/dt requires differentiating functions involving products of exponentials and trigonometric terms. Mastery of the product rule and derivatives of e^t, sin t, and cos t is necessary to correctly find these derivatives and proceed with the arc length calculation.
Recommended video:
6:04Introduction to Trigonometric Functions
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