Area of a Polar Region - AP Calculus BC
Card 1 of 30
State the symmetry about the polar axis condition for a polar curve.
State the symmetry about the polar axis condition for a polar curve.
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Replace $\theta$ with $-\theta$; identical equation. If curve unchanged when $\theta$ becomes $-\theta$.
Replace $\theta$ with $-\theta$; identical equation. If curve unchanged when $\theta$ becomes $-\theta$.
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What is the differential area element in polar coordinates?
What is the differential area element in polar coordinates?
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$dA = \frac{1}{2}r^2 d\theta$. Infinitesimal area element in polar form.
$dA = \frac{1}{2}r^2 d\theta$. Infinitesimal area element in polar form.
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What is the polar equation for a spiral of Archimedes?
What is the polar equation for a spiral of Archimedes?
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$r = a\theta$. Radius increases linearly with angle.
$r = a\theta$. Radius increases linearly with angle.
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State the form of a polar equation for a conic section.
State the form of a polar equation for a conic section.
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$r = \frac{ed}{1 + e\text{cos}\theta}$. General conic section in polar coordinates.
$r = \frac{ed}{1 + e\text{cos}\theta}$. General conic section in polar coordinates.
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Which polar equation represents a limaçon with an inner loop?
Which polar equation represents a limaçon with an inner loop?
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$r = a + b\text{cos}\theta$, $a < b$. When $a < b$, creates inner loop.
$r = a + b\text{cos}\theta$, $a < b$. When $a < b$, creates inner loop.
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State the symmetry about the line $\theta = \frac{\text{pi}}{2}$ for a polar curve.
State the symmetry about the line $\theta = \frac{\text{pi}}{2}$ for a polar curve.
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Replace $\theta$ with $\text{pi} - \theta$; identical equation. Reflection across the line $\theta = \frac{\pi}{2}$.
Replace $\theta$ with $\text{pi} - \theta$; identical equation. Reflection across the line $\theta = \frac{\pi}{2}$.
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What defines the symmetry about the origin for a polar curve?
What defines the symmetry about the origin for a polar curve?
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Replace $r$ with $-r$; identical equation. Point reflection through the origin.
Replace $r$ with $-r$; identical equation. Point reflection through the origin.
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What is the polar equation for a lemniscate?
What is the polar equation for a lemniscate?
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$r^2 = a^2 \text{cos}(2\theta)$. Figure-eight shaped curve.
$r^2 = a^2 \text{cos}(2\theta)$. Figure-eight shaped curve.
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What is the value of $\theta$ for a polar axis?
What is the value of $\theta$ for a polar axis?
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$\theta = 0$. The positive x-axis corresponds to $\theta = 0$.
$\theta = 0$. The positive x-axis corresponds to $\theta = 0$.
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Which polar function represents a cardioid?
Which polar function represents a cardioid?
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$r = a(1 + \text{cos}\theta)$. Heart-shaped curve with one cusp.
$r = a(1 + \text{cos}\theta)$. Heart-shaped curve with one cusp.
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Convert the polar point $(r, \theta)$ to Cartesian coordinates.
Convert the polar point $(r, \theta)$ to Cartesian coordinates.
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$(x, y) = (r\text{cos}\theta, r\text{sin}\theta)$. Standard polar to Cartesian conversion.
$(x, y) = (r\text{cos}\theta, r\text{sin}\theta)$. Standard polar to Cartesian conversion.
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State the condition for a polar curve having symmetry about the origin.
State the condition for a polar curve having symmetry about the origin.
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Replace $\theta$ with $\theta + \text{pi}$; identical equation. Alternative test for origin symmetry.
Replace $\theta$ with $\theta + \text{pi}$; identical equation. Alternative test for origin symmetry.
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Which polar equation represents a dimpled limaçon?
Which polar equation represents a dimpled limaçon?
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$r = a + b\text{cos}\theta$, $a > b$. When $a > b$, no inner loop forms.
$r = a + b\text{cos}\theta$, $a > b$. When $a > b$, no inner loop forms.
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What is the polar equation for a line through the pole?
What is the polar equation for a line through the pole?
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$\theta = \text{constant}$. Ray from origin at fixed angle.
$\theta = \text{constant}$. Ray from origin at fixed angle.
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Identify the polar equation for a circle centered at the origin with radius $a$.
Identify the polar equation for a circle centered at the origin with radius $a$.
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$r = a$. Constant radius from origin defines a circle.
$r = a$. Constant radius from origin defines a circle.
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Which polar equation represents a rose curve?
Which polar equation represents a rose curve?
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$r = a\text{cos}(n\theta)$ or $r = a\text{sin}(n\theta)$. Creates petals, number depends on $n$.
$r = a\text{cos}(n\theta)$ or $r = a\text{sin}(n\theta)$. Creates petals, number depends on $n$.
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Identify the polar equation for an ellipse.
Identify the polar equation for an ellipse.
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$r = \frac{a}{1 + e \cos \theta}$, $e < 1$. Conic with eccentricity less than 1.
$r = \frac{a}{1 + e \cos \theta}$, $e < 1$. Conic with eccentricity less than 1.
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What is the polar equation for a hyperbola?
What is the polar equation for a hyperbola?
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$r = \frac{a}{1 + e \cos \theta}$, $e > 1$. Conic with eccentricity greater than 1.
$r = \frac{a}{1 + e \cos \theta}$, $e > 1$. Conic with eccentricity greater than 1.
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What is the polar equation for a parabola?
What is the polar equation for a parabola?
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$r = \frac{a}{1 + e \cos \theta}$, $e = 1$. Conic with eccentricity exactly 1.
$r = \frac{a}{1 + e \cos \theta}$, $e = 1$. Conic with eccentricity exactly 1.
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Identify the range of $r$ for a polar region bounded by $r = 1 + \text{cos}\theta$.
Identify the range of $r$ for a polar region bounded by $r = 1 + \text{cos}\theta$.
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$0 \text{ to } 2$. Cardioid ranges from minimum 0 to maximum 2.
$0 \text{ to } 2$. Cardioid ranges from minimum 0 to maximum 2.
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State the formula for converting angular coordinates to polar.
State the formula for converting angular coordinates to polar.
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$\theta = \text{tan}^{-1}(\frac{y}{x})$. Angular coordinate conversion from Cartesian.
$\theta = \text{tan}^{-1}(\frac{y}{x})$. Angular coordinate conversion from Cartesian.
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Convert Cartesian coordinates $(x, y)$ to polar coordinates.
Convert Cartesian coordinates $(x, y)$ to polar coordinates.
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$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Standard Cartesian to polar conversion.
$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Standard Cartesian to polar conversion.
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Which polar equation represents a limaçon without an inner loop?
Which polar equation represents a limaçon without an inner loop?
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$r = a + b\text{cos}\theta$, $a = b$. Boundary case between loop and no loop.
$r = a + b\text{cos}\theta$, $a = b$. Boundary case between loop and no loop.
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Which polar equation represents a limaçon without an inner loop?
Which polar equation represents a limaçon without an inner loop?
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$r = a + b\text{cos}\theta$, $a = b$. Boundary case between loop and no loop.
$r = a + b\text{cos}\theta$, $a = b$. Boundary case between loop and no loop.
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Convert Cartesian coordinates $(x, y)$ to polar coordinates.
Convert Cartesian coordinates $(x, y)$ to polar coordinates.
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$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Standard Cartesian to polar conversion.
$(r, \theta) = (\text{sqrt}(x^2 + y^2), \text{tan}^{-1}(\frac{y}{x}))$. Standard Cartesian to polar conversion.
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State the formula for converting angular coordinates to polar.
State the formula for converting angular coordinates to polar.
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$\theta = \text{tan}^{-1}(\frac{y}{x})$. Angular coordinate conversion from Cartesian.
$\theta = \text{tan}^{-1}(\frac{y}{x})$. Angular coordinate conversion from Cartesian.
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Identify the range of $r$ for a polar region bounded by $r = 1 + \text{cos}\theta$.
Identify the range of $r$ for a polar region bounded by $r = 1 + \text{cos}\theta$.
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$0 \text{ to } 2$. Cardioid ranges from minimum 0 to maximum 2.
$0 \text{ to } 2$. Cardioid ranges from minimum 0 to maximum 2.
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What is the polar equation for a parabola?
What is the polar equation for a parabola?
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$r = \frac{a}{1 + e\text{cos}\theta}$, $e = 1$. Conic with eccentricity exactly 1.
$r = \frac{a}{1 + e\text{cos}\theta}$, $e = 1$. Conic with eccentricity exactly 1.
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What defines the symmetry about the origin for a polar curve?
What defines the symmetry about the origin for a polar curve?
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Replace $r$ with $-r$; identical equation. Point reflection through the origin.
Replace $r$ with $-r$; identical equation. Point reflection through the origin.
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State the symmetry about the line $\theta = \frac{\text{pi}}{2}$ for a polar curve.
State the symmetry about the line $\theta = \frac{\text{pi}}{2}$ for a polar curve.
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Replace $\theta$ with $\text{pi} - \theta$; identical equation. Reflection across the line $\theta = \frac{\pi}{2}$.
Replace $\theta$ with $\text{pi} - \theta$; identical equation. Reflection across the line $\theta = \frac{\pi}{2}$.
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