Area Bounded By Two Polar Curves - AP Calculus BC
Card 1 of 30
Calculate the complete area enclosed by $r = 2\cos(\theta)$.
Calculate the complete area enclosed by $r = 2\cos(\theta)$.
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$A = \frac{1}{2} \int_{0}^{2\pi} (2\cos(\theta))^2 , d\theta$. Circle area formula integrated over full period.
$A = \frac{1}{2} \int_{0}^{2\pi} (2\cos(\theta))^2 , d\theta$. Circle area formula integrated over full period.
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Determine the area between $r = 1 + \sin(\theta)$ and $r = 1$ for $\theta = 0$ to $\theta = \pi$.
Determine the area between $r = 1 + \sin(\theta)$ and $r = 1$ for $\theta = 0$ to $\theta = \pi$.
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$\frac{1}{2} \int_{0}^{\pi} ((1 + \sin(\theta))^2 - 1) , d\theta$. Cardioid minus circle area using difference formula.
$\frac{1}{2} \int_{0}^{\pi} ((1 + \sin(\theta))^2 - 1) , d\theta$. Cardioid minus circle area using difference formula.
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What is the integral expression for the area inside $r = 3\sin(\theta)$ but outside $r = 1$?
What is the integral expression for the area inside $r = 3\sin(\theta)$ but outside $r = 1$?
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$\frac{1}{2} \int_{\alpha}^{\beta} (9\sin^2(\theta) - 1) , d\theta$. Region between circle and line using difference formula.
$\frac{1}{2} \int_{\alpha}^{\beta} (9\sin^2(\theta) - 1) , d\theta$. Region between circle and line using difference formula.
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What is the role of symmetry in simplifying polar area calculations?
What is the role of symmetry in simplifying polar area calculations?
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Allows reducing integration limits. Reduces computational work by exploiting curve symmetries.
Allows reducing integration limits. Reduces computational work by exploiting curve symmetries.
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What does $d\theta$ represent in the polar area integral?
What does $d\theta$ represent in the polar area integral?
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A small change in the angle $\theta$. Represents an infinitesimal angular increment in polar coordinates.
A small change in the angle $\theta$. Represents an infinitesimal angular increment in polar coordinates.
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What does the expression $r = a(1 \pm \cos(\theta))$ represent?
What does the expression $r = a(1 \pm \cos(\theta))$ represent?
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A cardioid. Heart-shaped curve created when $a = b$ in limaçon equation.
A cardioid. Heart-shaped curve created when $a = b$ in limaçon equation.
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Determine the area between $r = 3\cos(\theta)$ and $r = 1$ from $\theta = 0$ to $\theta = \pi$.
Determine the area between $r = 3\cos(\theta)$ and $r = 1$ from $\theta = 0$ to $\theta = \pi$.
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$\frac{1}{2} \int_{0}^{\pi} (9\cos^2(\theta) - 1) , d\theta$. Circle minus circle area using difference formula.
$\frac{1}{2} \int_{0}^{\pi} (9\cos^2(\theta) - 1) , d\theta$. Circle minus circle area using difference formula.
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What is the polar area formula for the circle $r = a \cos(\theta)$?
What is the polar area formula for the circle $r = a \cos(\theta)$?
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$A = \frac{1}{2} \int_{0}^{\pi} (a\cos(\theta))^2 , d\theta$. Circle formula integrated over half period.
$A = \frac{1}{2} \int_{0}^{\pi} (a\cos(\theta))^2 , d\theta$. Circle formula integrated over half period.
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Determine which curve is outer: $r = 3 + 2\sin(\theta)$ or $r = 2$?
Determine which curve is outer: $r = 3 + 2\sin(\theta)$ or $r = 2$?
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$r = 3 + 2\sin(\theta)$ is outer. Limaçon has larger radius values than the constant circle.
$r = 3 + 2\sin(\theta)$ is outer. Limaçon has larger radius values than the constant circle.
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State the formula for the area between two polar curves $r = f(\theta)$ and $r = g(\theta)$.
State the formula for the area between two polar curves $r = f(\theta)$ and $r = g(\theta)$.
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$A = \frac{1}{2} \int_{\alpha}^{\beta} (f(\theta)^2 - g(\theta)^2) , d\theta$. Subtracts the inner curve's area from the outer curve's area.
$A = \frac{1}{2} \int_{\alpha}^{\beta} (f(\theta)^2 - g(\theta)^2) , d\theta$. Subtracts the inner curve's area from the outer curve's area.
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Identify the correct integration bounds for a full polar circle $r = a$.
Identify the correct integration bounds for a full polar circle $r = a$.
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$\theta = 0$ to $\theta = 2\pi$. A complete circle requires one full rotation around the origin.
$\theta = 0$ to $\theta = 2\pi$. A complete circle requires one full rotation around the origin.
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What is the area of the region enclosed by $r = 2 + 2\cos(\theta)$?
What is the area of the region enclosed by $r = 2 + 2\cos(\theta)$?
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$A = \frac{1}{2} \int_{0}^{2\pi} (2 + 2\cos(\theta))^2 , d\theta$. Cardioid area formula integrated over full period.
$A = \frac{1}{2} \int_{0}^{2\pi} (2 + 2\cos(\theta))^2 , d\theta$. Cardioid area formula integrated over full period.
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How do you convert a polar area integral to Cartesian form?
How do you convert a polar area integral to Cartesian form?
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Use $x = r\cos(\theta)$, $y = r\sin(\theta)$. Uses standard polar-to-Cartesian coordinate transformation.
Use $x = r\cos(\theta)$, $y = r\sin(\theta)$. Uses standard polar-to-Cartesian coordinate transformation.
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State the formula to convert $r = f(\theta)$ to Cartesian coordinates.
State the formula to convert $r = f(\theta)$ to Cartesian coordinates.
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$x = r\cos(\theta)$, $y = r\sin(\theta)$. Standard polar-to-Cartesian transformation formulas.
$x = r\cos(\theta)$, $y = r\sin(\theta)$. Standard polar-to-Cartesian transformation formulas.
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Identify the area enclosed by $r = 1 - \cos(\theta)$ from $\theta = 0$ to $\theta = \pi$.
Identify the area enclosed by $r = 1 - \cos(\theta)$ from $\theta = 0$ to $\theta = \pi$.
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$A = \frac{1}{2} \int_{0}^{\pi} (1 - \cos(\theta))^2 , d\theta$. Cardioid area formula integrated over half period.
$A = \frac{1}{2} \int_{0}^{\pi} (1 - \cos(\theta))^2 , d\theta$. Cardioid area formula integrated over half period.
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What is the area of one petal of the rose curve $r = 2\sin(2\theta)$?
What is the area of one petal of the rose curve $r = 2\sin(2\theta)$?
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$\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (2\sin(2\theta))^2 , d\theta$. Four-petal rose with area of one petal calculated.
$\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (2\sin(2\theta))^2 , d\theta$. Four-petal rose with area of one petal calculated.
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For $r = 2 + \cos(\theta)$, what is the maximum value of $r$?
For $r = 2 + \cos(\theta)$, what is the maximum value of $r$?
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$r = 3$. Maximum occurs when $\cos(\theta) = 1$, so $r = 2 + 1 = 3$.
$r = 3$. Maximum occurs when $\cos(\theta) = 1$, so $r = 2 + 1 = 3$.
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What is the area enclosed by $r = 2 + 3\cos(\theta)$ from $\theta = 0$ to $\theta = \pi$?
What is the area enclosed by $r = 2 + 3\cos(\theta)$ from $\theta = 0$ to $\theta = \pi$?
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$\frac{1}{2} \int_{0}^{\pi} (2 + 3\cos(\theta))^2 , d\theta$. Limaçon area formula integrated over half period.
$\frac{1}{2} \int_{0}^{\pi} (2 + 3\cos(\theta))^2 , d\theta$. Limaçon area formula integrated over half period.
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Identify the region type for $r = a \pm b\cos(\theta)$ when $a < b$.
Identify the region type for $r = a \pm b\cos(\theta)$ when $a < b$.
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Limaçon with an inner loop. When inner coefficient exceeds outer, creates inner loop.
Limaçon with an inner loop. When inner coefficient exceeds outer, creates inner loop.
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Identify the area enclosed by $r = 2 - 2\sin(\theta)$ from $\theta = 0$ to $\theta = 2\pi$.
Identify the area enclosed by $r = 2 - 2\sin(\theta)$ from $\theta = 0$ to $\theta = 2\pi$.
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$A = \frac{1}{2} \int_{0}^{2\pi} (2 - 2\sin(\theta))^2 , d\theta$. Cardioid area formula integrated over full period.
$A = \frac{1}{2} \int_{0}^{2\pi} (2 - 2\sin(\theta))^2 , d\theta$. Cardioid area formula integrated over full period.
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Which polar curve represents a cardioid: $r = 1 + \cos(\theta)$ or $r = 2\sin(\theta)$?
Which polar curve represents a cardioid: $r = 1 + \cos(\theta)$ or $r = 2\sin(\theta)$?
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$r = 1 + \cos(\theta)$. Heart-shaped curve with cusp at origin.
$r = 1 + \cos(\theta)$. Heart-shaped curve with cusp at origin.
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Which function describes a limaçon: $r = a \pm b\sin(\theta)$ or $r = a\cos(\theta)$?
Which function describes a limaçon: $r = a \pm b\sin(\theta)$ or $r = a\cos(\theta)$?
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$r = a \pm b\sin(\theta)$. Limaçon equation includes both sine and cosine variations.
$r = a \pm b\sin(\theta)$. Limaçon equation includes both sine and cosine variations.
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Find the area inside $r = 4\cos(\theta)$ but outside $r = 2$.
Find the area inside $r = 4\cos(\theta)$ but outside $r = 2$.
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$\frac{1}{2} \int_{\alpha}^{\beta} (16\cos^2(\theta) - 4) , d\theta$. Circle minus line area using difference formula.
$\frac{1}{2} \int_{\alpha}^{\beta} (16\cos^2(\theta) - 4) , d\theta$. Circle minus line area using difference formula.
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What is the result of $\int_{0}^{\pi} \sin^2(\theta) , d\theta$?
What is the result of $\int_{0}^{\pi} \sin^2(\theta) , d\theta$?
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$\frac{\pi}{2}$. Uses the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$.
$\frac{\pi}{2}$. Uses the identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$.
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Choose the expression that represents the area of a single polar curve $r = f(\theta)$.
Choose the expression that represents the area of a single polar curve $r = f(\theta)$.
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$A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 , d\theta$. Standard formula for area enclosed by one polar curve.
$A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 , d\theta$. Standard formula for area enclosed by one polar curve.
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State the general steps to calculate the area between two polar curves.
State the general steps to calculate the area between two polar curves.
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Find intersections, set up integral, evaluate. Standard procedure: intersections, integral setup, evaluation.
Find intersections, set up integral, evaluate. Standard procedure: intersections, integral setup, evaluation.
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How do you find the points of intersection of two polar curves $r = f(\theta)$ and $r = g(\theta)$?
How do you find the points of intersection of two polar curves $r = f(\theta)$ and $r = g(\theta)$?
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Solve $f(\theta) = g(\theta)$ for $\theta$. Sets equal the radial distances to find where curves meet.
Solve $f(\theta) = g(\theta)$ for $\theta$. Sets equal the radial distances to find where curves meet.
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What is the polar area formula if $g(\theta) = 0$?
What is the polar area formula if $g(\theta) = 0$?
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$A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 , d\theta$. Reduces to the single curve area formula when inner curve is zero.
$A = \frac{1}{2} \int_{\alpha}^{\beta} f(\theta)^2 , d\theta$. Reduces to the single curve area formula when inner curve is zero.
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Identify the integral bounds when finding the area between $r = 2\cos(\theta)$ and $r = 1$.
Identify the integral bounds when finding the area between $r = 2\cos(\theta)$ and $r = 1$.
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$\theta = 0$ to $\theta = \frac{\pi}{3}$. Found by solving $2\cos(\theta) = 1$ for intersection points.
$\theta = 0$ to $\theta = \frac{\pi}{3}$. Found by solving $2\cos(\theta) = 1$ for intersection points.
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Calculate the area between $r = 3\sin(\theta)$ and $r = 2$ from $\theta = 0$ to $\theta = \frac{\pi}{2}$.
Calculate the area between $r = 3\sin(\theta)$ and $r = 2$ from $\theta = 0$ to $\theta = \frac{\pi}{2}$.
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$\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (9\sin^2(\theta) - 4) , d\theta$. Uses the difference formula with $f(\theta) = 3\sin(\theta)$ and $g(\theta) = 2$.
$\frac{1}{2} \int_{0}^{\frac{\pi}{2}} (9\sin^2(\theta) - 4) , d\theta$. Uses the difference formula with $f(\theta) = 3\sin(\theta)$ and $g(\theta) = 2$.
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