교과서 질문
Length in Polar Coordinates
Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.
r = √(1 + cos 2θ), −π/2 ≤ θ ≤ π/2
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
11장, 문제 11.PE.33
Cartesian to Polar Equations
Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.
x² + y² + 5y = 0
검증된 단계별 안내1
Start with the given Cartesian equation of the circle: \(x^{2} + y^{2} + 5y = 0\).
Rewrite the equation to complete the square for the \(y\) terms. Group \(y^{2} + 5y\) together: \(x^{2} + (y^{2} + 5y) = 0\).
Complete the square for \(y^{2} + 5y\) by adding and subtracting \(\left(\frac{5}{2}\right)^{2} = \frac{25}{4}\) inside the equation: \(x^{2} + \left(y^{2} + 5y + \frac{25}{4}\right) - \frac{25}{4} = 0\).
Rewrite the equation as \(x^{2} + \left(y + \frac{5}{2}\right)^{2} = \frac{25}{4}\), which represents a circle with center at \((0, -\frac{5}{2})\) and radius \(\frac{5}{2}\).
Convert to polar coordinates using \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\). Substitute these into the equation: \(\left(r \cos{\theta}\right)^{2} + \left(r \sin{\theta} + \frac{5}{2}\right)^{2} = \frac{25}{4}\). This is the polar form of the circle's equation.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
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Conversion between Cartesian and Polar Coordinates
This concept involves expressing points from the Cartesian coordinate system (x, y) in terms of polar coordinates (r, θ), where x = r cos θ and y = r sin θ. Understanding this conversion is essential to rewrite Cartesian equations in polar form.
추천 영상:
05:32
Intro to Polar Coordinates
Equation of a Circle in Cartesian Coordinates
A circle in Cartesian form is typically expressed as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Recognizing and rewriting the given equation into this standard form helps identify the circle’s center and radius.
추천 영상:
05:32Intro to Polar Coordinates
Polar Form of a Circle Equation
After converting Cartesian coordinates to polar, the circle’s equation can be expressed in terms of r and θ. This often involves substituting x = r cos θ and y = r sin θ and simplifying to isolate r, which helps in sketching and understanding the circle in polar coordinates.
추천 영상:
6:50Convert Equations from Polar to Rectangular
관련 실천
교과서 질문
Graphing Conic Sections
Sketch the parabolas in Exercises 55–58. Include the focus and directrix in each sketch.
y² = −(8/3)x
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교과서 질문
Lengths of Curves
Find the lengths of the curves in Exercises 13–19.
x = 5 cos t − cos 5t, y = 5 sin t − sin 5t, 0 ≤ t ≤ π/2
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교과서 질문
Finding Parametric Equations
Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².
a. once clockwise.
(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)
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교과서 질문
Area in Polar Coordinates
Find the areas of the regions in the polar coordinate plane described in Exercises 47–50.
Inside the cardioid r = 2(1 + sin θ) and outside the circle r = 2 sin θ
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교과서 질문
Cycloid
a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).
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