Area Between Curves: Functions of y - AP Calculus AB
Card 1 of 30
Find area between $x = 4 - y^2$ and $x = y^2$ from $y = -2$ to $y = 2$.
Find area between $x = 4 - y^2$ and $x = y^2$ from $y = -2$ to $y = 2$.
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$\text{integral of } (4 - 2y^2) \text{ dy}$. Since $4-y^2 > y^2$ when $y^2 < 2$, integrate $(4-2y^2)$.
$\text{integral of } (4 - 2y^2) \text{ dy}$. Since $4-y^2 > y^2$ when $y^2 < 2$, integrate $(4-2y^2)$.
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Convert $y = x^2$ and $x = 4$ to functions of $y$ for integration.
Convert $y = x^2$ and $x = 4$ to functions of $y$ for integration.
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$x = \text{sqrt}(y)$ and $x = 4$. Solve $y=x^2$ for $x$ to get $x=\sqrt{y}$; keep $x=4$.
$x = \text{sqrt}(y)$ and $x = 4$. Solve $y=x^2$ for $x$ to get $x=\sqrt{y}$; keep $x=4$.
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Calculate intersection points for $x = y^2$ and $x = 4 - y^2$.
Calculate intersection points for $x = y^2$ and $x = 4 - y^2$.
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Solve $y^2 = 4 - y^2$. Set functions equal: $y^2 = 4-y^2$ gives $2y^2 = 4$.
Solve $y^2 = 4 - y^2$. Set functions equal: $y^2 = 4-y^2$ gives $2y^2 = 4$.
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What is the condition for $x = f(y)$ to be left of $x = g(y)$?
What is the condition for $x = f(y)$ to be left of $x = g(y)$?
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$f(y) < g(y)$. For curves expressed as $x=f(y)$ and $x=g(y)$.
$f(y) < g(y)$. For curves expressed as $x=f(y)$ and $x=g(y)$.
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What is $g(y)$ for the curve $x = y^2 + 1$?
What is $g(y)$ for the curve $x = y^2 + 1$?
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$g(y) = y^2 + 1$. Direct identification of the function from the equation.
$g(y) = y^2 + 1$. Direct identification of the function from the equation.
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Find the area between $x = y^2$ and $x = 4$ from $y = -2$ to $y = 2$.
Find the area between $x = y^2$ and $x = 4$ from $y = -2$ to $y = 2$.
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$\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ for $y \in [-2,2]$, integrate $(4-y^2)$.
$\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ for $y \in [-2,2]$, integrate $(4-y^2)$.
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State the integral for area between $x = y^2 + y$ and $x = y + 3$.
State the integral for area between $x = y^2 + y$ and $x = y + 3$.
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$\text{integral of } (y + 3 - (y^2 + y)) \text{ dy}$. Simplifies to integral of $(3-y^2)$ after expanding.
$\text{integral of } (y + 3 - (y^2 + y)) \text{ dy}$. Simplifies to integral of $(3-y^2)$ after expanding.
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State the formula for area between $x = f(y)$ and $x = g(y)$ using integration.
State the formula for area between $x = f(y)$ and $x = g(y)$ using integration.
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$\text{Area} = \text{integral of } [f(y) - g(y)] \text{ dy}$. General area formula between two curves expressed as functions of $y$.
$\text{Area} = \text{integral of } [f(y) - g(y)] \text{ dy}$. General area formula between two curves expressed as functions of $y$.
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Find points of intersection for $x = y^2$ and $x = 2 - y$.
Find points of intersection for $x = y^2$ and $x = 2 - y$.
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Solve $y^2 = 2 - y$. Set functions equal to find intersection $y$-coordinates.
Solve $y^2 = 2 - y$. Set functions equal to find intersection $y$-coordinates.
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Find the intersection points of $x = y^2 - 3$ and $x = 5 - y$.
Find the intersection points of $x = y^2 - 3$ and $x = 5 - y$.
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Solve $y^2 - 3 = 5 - y$. Set the functions equal to find intersection points.
Solve $y^2 - 3 = 5 - y$. Set the functions equal to find intersection points.
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Identify $y$-limits for $x = y^2 + 1$ and $x = y + 3$.
Identify $y$-limits for $x = y^2 + 1$ and $x = y + 3$.
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Solve $y^2 + 1 = y + 3$. Set functions equal to find $y$-intersection points.
Solve $y^2 + 1 = y + 3$. Set functions equal to find $y$-intersection points.
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What must be true about limits $y_1$ and $y_2$ for integration?
What must be true about limits $y_1$ and $y_2$ for integration?
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$y_1 < y_2$. Lower limit must be less than upper limit for proper integration.
$y_1 < y_2$. Lower limit must be less than upper limit for proper integration.
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State the condition for $f(y)$ and $g(y)$ to find area between curves.
State the condition for $f(y)$ and $g(y)$ to find area between curves.
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$f(y) \text{ should be greater than } g(y) \text{ in the interval}$. Ensures $f(y) - g(y) \geq 0$ for positive area calculation.
$f(y) \text{ should be greater than } g(y) \text{ in the interval}$. Ensures $f(y) - g(y) \geq 0$ for positive area calculation.
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Determine area between $x = 4$ and $x = y^2$ from $y = -2$ to $y = 2$.
Determine area between $x = 4$ and $x = y^2$ from $y = -2$ to $y = 2$.
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$\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Same setup as flashcard 3: $4 > y^2$ on the interval.
$\text{Area} = \text{integral from } -2 \text{ to } 2 \text{ of } (4 - y^2) \text{ dy}$. Same setup as flashcard 3: $4 > y^2$ on the interval.
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What does the integral of $[f(y) - g(y)]$ represent when $x = f(y)$ and $x = g(y)$?
What does the integral of $[f(y) - g(y)]$ represent when $x = f(y)$ and $x = g(y)$?
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Area between the curves. The definite integral gives the signed area between curves.
Area between the curves. The definite integral gives the signed area between curves.
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Determine area between $x = y^2$ and $x = 4$ from $y = 0$ to $y = 2$.
Determine area between $x = y^2$ and $x = 4$ from $y = 0$ to $y = 2$.
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$\text{integral of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ on $[0,2]$, integrate $(4-y^2)$.
$\text{integral of } (4 - y^2) \text{ dy}$. Since $4 > y^2$ on $[0,2]$, integrate $(4-y^2)$.
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Identify $y_1$ and $y_2$ for $x = 3y$ and $x = y^2$ if intersecting at $y = 0$ and $y = 3$.
Identify $y_1$ and $y_2$ for $x = 3y$ and $x = y^2$ if intersecting at $y = 0$ and $y = 3$.
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$y_1 = 0, y_2 = 3$. These are the given intersection $y$-coordinates for the curves.
$y_1 = 0, y_2 = 3$. These are the given intersection $y$-coordinates for the curves.
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Which function is rightmost: $x = y^2$ or $x = 3$?
Which function is rightmost: $x = y^2$ or $x = 3$?
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$x = 3$. Constant function $x=3$ is always to the right of $x=y^2$.
$x = 3$. Constant function $x=3$ is always to the right of $x=y^2$.
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Find the intersection points for $x = 3y - y^2$ and $x = y$.
Find the intersection points for $x = 3y - y^2$ and $x = y$.
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Solve $3y - y^2 = y$. Set functions equal to find where curves intersect.
Solve $3y - y^2 = y$. Set functions equal to find where curves intersect.
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Determine $f(y)$ and $g(y)$ for $x = y^2 + 1$ and $x = y + 3$.
Determine $f(y)$ and $g(y)$ for $x = y^2 + 1$ and $x = y + 3$.
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$f(y) = y + 3, g(y) = y^2 + 1$. Identify which function is rightmost to determine $f(y)$ and $g(y)$.
$f(y) = y + 3, g(y) = y^2 + 1$. Identify which function is rightmost to determine $f(y)$ and $g(y)$.
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Which function is upper: $x = y + 1$ or $x = y^2$ for $y \text{ in } [0, 1]$?
Which function is upper: $x = y + 1$ or $x = y^2$ for $y \text{ in } [0, 1]$?
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$x = y + 1$. For $y \in [0,1]$: $y+1 > y^2$ since line above parabola.
$x = y + 1$. For $y \in [0,1]$: $y+1 > y^2$ since line above parabola.
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Find the limits of integration for $x = y + 1$ and $x = -y + 3$.
Find the limits of integration for $x = y + 1$ and $x = -y + 3$.
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Solve $y + 1 = -y + 3$. Set the functions equal to solve for intersection $y$-values.
Solve $y + 1 = -y + 3$. Set the functions equal to solve for intersection $y$-values.
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What is the role of intersection points in finding areas?
What is the role of intersection points in finding areas?
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They determine limits of integration. Intersections provide the integration bounds for area calculation.
They determine limits of integration. Intersections provide the integration bounds for area calculation.
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Find intersections of $x = 2y$ and $x = y^2$.
Find intersections of $x = 2y$ and $x = y^2$.
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Solve $2y = y^2$. Set the functions equal: $2y = y^2$ to find intersections.
Solve $2y = y^2$. Set the functions equal: $2y = y^2$ to find intersections.
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Which function is upper: $x = y^2$ or $x = 4$ for $y \text{ in } [-2, 2]$?
Which function is upper: $x = y^2$ or $x = 4$ for $y \text{ in } [-2, 2]$?
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$x = 4$. Constant function $x=4$ is always to the right of $x=y^2$.
$x = 4$. Constant function $x=4$ is always to the right of $x=y^2$.
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For curves $x = y^3$ and $x = 2y$, identify $y_1$ and $y_2$ if they intersect at $y = -1$ and $y = 2$.
For curves $x = y^3$ and $x = 2y$, identify $y_1$ and $y_2$ if they intersect at $y = -1$ and $y = 2$.
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$y_1 = -1, y_2 = 2$. These are the $y$-coordinates where the curves intersect.
$y_1 = -1, y_2 = 2$. These are the $y$-coordinates where the curves intersect.
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Identify the region of integration for curves $x = f(y)$ and $x = g(y)$ between $y = a$ and $y = b$.
Identify the region of integration for curves $x = f(y)$ and $x = g(y)$ between $y = a$ and $y = b$.
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From $y = a$ to $y = b$. Integration bounds for functions of $y$ are the $y$-values.
From $y = a$ to $y = b$. Integration bounds for functions of $y$ are the $y$-values.
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What is the formula for finding area between two curves $x = f(y)$ and $x = g(y)$?
What is the formula for finding area between two curves $x = f(y)$ and $x = g(y)$?
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$\text{Area} = \int_{y_1}^{y_2} [f(y) - g(y)] , \text{dy}$. Standard formula where $f(y) > g(y)$ for the entire interval.
$\text{Area} = \int_{y_1}^{y_2} [f(y) - g(y)] , \text{dy}$. Standard formula where $f(y) > g(y)$ for the entire interval.
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Find the area between $x = 3y$ and $x = y^2$ from $y = 0$ to $y = 3$.
Find the area between $x = 3y$ and $x = y^2$ from $y = 0$ to $y = 3$.
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$$\text{Area} = \int_0^3 (3y - y^2) , dy$$. Since $3y > y^2$ on $[0,3]$, integrate $(3y-y^2)$.
$$\text{Area} = \int_0^3 (3y - y^2) , dy$$. Since $3y > y^2$ on $[0,3]$, integrate $(3y-y^2)$.
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What is the integral expression for area between $x = y$ and $x = y^2$?
What is the integral expression for area between $x = y$ and $x = y^2$?
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$\text{integral of } (y - y^2) \text{ dy}$. Since $y > y^2$ on $(0,1)$, integrate $(y-y^2)$.
$\text{integral of } (y - y^2) \text{ dy}$. Since $y > y^2$ on $(0,1)$, integrate $(y-y^2)$.
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