Area Between Curves with Multiple Intersections - AP Calculus AB
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When should you re-evaluate curve positions in an area problem?
When should you re-evaluate curve positions in an area problem?
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If the curves intersect within the interval. Intersections change which curve is on top.
If the curves intersect within the interval. Intersections change which curve is on top.
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Determine the area between $y = x^2$ and $y = 4 - x^2$ on $[-2, 2]$.
Determine the area between $y = x^2$ and $y = 4 - x^2$ on $[-2, 2]$.
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$A = 2 \int_0^2 (4 - 2x^2) , dx$. Uses symmetry since $4 - x^2 > x^2$ on $[-2, 2]$.
$A = 2 \int_0^2 (4 - 2x^2) , dx$. Uses symmetry since $4 - x^2 > x^2$ on $[-2, 2]$.
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What integral calculates the area between $y = x^2$ and $y = x$ on $[0, 1]$?
What integral calculates the area between $y = x^2$ and $y = x$ on $[0, 1]$?
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$A = \int_0^1 (x - x^2) , dx$. $y = x$ is above $y = x^2$ on $[0, 1]$.
$A = \int_0^1 (x - x^2) , dx$. $y = x$ is above $y = x^2$ on $[0, 1]$.
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Which method can be used if curves intersect more than twice?
Which method can be used if curves intersect more than twice?
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Split the integration into regions based on intersections. Each segment has different upper/lower curves.
Split the integration into regions based on intersections. Each segment has different upper/lower curves.
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Determine the intersection points of $y = x^3$ and $y = x$.
Determine the intersection points of $y = x^3$ and $y = x$.
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Intersection points: $(0, 0)$ and $(1, 1)$. Solve $x^3 = x$ to find where curves meet.
Intersection points: $(0, 0)$ and $(1, 1)$. Solve $x^3 = x$ to find where curves meet.
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Find the intersection points of $y = x$ and $y = x^2$.
Find the intersection points of $y = x$ and $y = x^2$.
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Intersection points: $(0, 0)$ and $(1, 1)$. Set $x = x^2$ and solve for intersection points.
Intersection points: $(0, 0)$ and $(1, 1)$. Set $x = x^2$ and solve for intersection points.
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What happens when $f(x)$ and $g(x)$ switch positions in the interval?
What happens when $f(x)$ and $g(x)$ switch positions in the interval?
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Adjust the integrals to reflect the change in position. Swap the order in the integrand appropriately.
Adjust the integrals to reflect the change in position. Swap the order in the integrand appropriately.
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Write the integral for the area between $y = x$ and $y = x^3$ on $[0, 1]$.
Write the integral for the area between $y = x$ and $y = x^3$ on $[0, 1]$.
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$A = \int_0^1 (x - x^3) , dx$. $y = x$ is above $y = x^3$ on $[0, 1]$.
$A = \int_0^1 (x - x^3) , dx$. $y = x$ is above $y = x^3$ on $[0, 1]$.
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What do you do if curves intersect at more than two points?
What do you do if curves intersect at more than two points?
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Split into separate integrals for each segment. Each segment needs its own integral calculation.
Split into separate integrals for each segment. Each segment needs its own integral calculation.
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Calculate the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$.
Calculate the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$.
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$A = \int_0^3 (3x - x^2) , dx$. $3x$ is above $x^2$ on this interval.
$A = \int_0^3 (3x - x^2) , dx$. $3x$ is above $x^2$ on this interval.
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What is the integral setup for $y = 4x - x^2$ and $y = x^2$ on $[0, 2]$?
What is the integral setup for $y = 4x - x^2$ and $y = x^2$ on $[0, 2]$?
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$A = \int_0^2 (4x - 2x^2) , dx$. $4x - x^2$ is above $x^2$ on this interval.
$A = \int_0^2 (4x - 2x^2) , dx$. $4x - x^2$ is above $x^2$ on this interval.
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What is the area between $y = e^x$ and $y = 1$ from $x = 0$ to $x = 1$?
What is the area between $y = e^x$ and $y = 1$ from $x = 0$ to $x = 1$?
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$A = \int_0^1 (e^x - 1) , dx$. $e^x > 1$ for all $x > 0$.
$A = \int_0^1 (e^x - 1) , dx$. $e^x > 1$ for all $x > 0$.
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Calculate the area between $y = x^3$ and $y = 2x^2$ from $x = 0$ to $x = 2$.
Calculate the area between $y = x^3$ and $y = 2x^2$ from $x = 0$ to $x = 2$.
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$A = \int_0^2 (2x^2 - x^3) , dx$. $2x^2 > x^3$ on this interval.
$A = \int_0^2 (2x^2 - x^3) , dx$. $2x^2 > x^3$ on this interval.
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What are often the limits of integration for area problems?
What are often the limits of integration for area problems?
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The x-coordinates of the intersections of the curves. Intersections naturally define integration bounds.
The x-coordinates of the intersections of the curves. Intersections naturally define integration bounds.
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Identify the integral for $y = x^2$ and $y = 5 - x^2$ on $[0, \sqrt{5}]$.
Identify the integral for $y = x^2$ and $y = 5 - x^2$ on $[0, \sqrt{5}]$.
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$A = \int_0^{\sqrt{5}} (5 - 2x^2) , dx$. $5 - x^2 > x^2$ gives $5 > 2x^2$.
$A = \int_0^{\sqrt{5}} (5 - 2x^2) , dx$. $5 - x^2 > x^2$ gives $5 > 2x^2$.
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What is the formula for calculating area between curves if $f(x)$ and $g(x)$ switch?
What is the formula for calculating area between curves if $f(x)$ and $g(x)$ switch?
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Separate integrals for each segment with correct top curve. Account for position changes at intersections.
Separate integrals for each segment with correct top curve. Account for position changes at intersections.
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For $y = 2x$ and $y = x^3$, what is the area on $[0, 2]$?
For $y = 2x$ and $y = x^3$, what is the area on $[0, 2]$?
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$A = \int_0^2 (2x - x^3) , dx$. $2x > x^3$ on this interval.
$A = \int_0^2 (2x - x^3) , dx$. $2x > x^3$ on this interval.
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What is the integral setup for the area between $y = 4x$ and $y = x^3$ on $[0, 2]$?
What is the integral setup for the area between $y = 4x$ and $y = x^3$ on $[0, 2]$?
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$A = \int_0^2 (4x - x^3) , dx$. $4x > x^3$ on this interval.
$A = \int_0^2 (4x - x^3) , dx$. $4x > x^3$ on this interval.
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What is the next step after finding intersections for area calculation?
What is the next step after finding intersections for area calculation?
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Determine which curve is above the other in each interval. Test points reveal which function is greater.
Determine which curve is above the other in each interval. Test points reveal which function is greater.
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What step determines the integration limits for the area between curves?
What step determines the integration limits for the area between curves?
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Find the points of intersection of the curves. Intersections define where curves switch position.
Find the points of intersection of the curves. Intersections define where curves switch position.
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Find the area between $y = 2x$ and $y = x^2$ from $x = 0$ to $x = 2$.
Find the area between $y = 2x$ and $y = x^2$ from $x = 0$ to $x = 2$.
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$A = \int_0^2 (2x - x^2) , dx$. $2x$ is above $x^2$ on this interval.
$A = \int_0^2 (2x - x^2) , dx$. $2x$ is above $x^2$ on this interval.
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For $y = x^2$ and $y = 2x$, what is the area on $[0, 2]$?
For $y = x^2$ and $y = 2x$, what is the area on $[0, 2]$?
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$A = \int_0^2 (2x - x^2) , dx$. $2x > x^2$ on this interval.
$A = \int_0^2 (2x - x^2) , dx$. $2x > x^2$ on this interval.
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What is the area between $y = 3x^2$ and $y = x^3$ from $x = 0$ to $x = 3$?
What is the area between $y = 3x^2$ and $y = x^3$ from $x = 0$ to $x = 3$?
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$A = \int_0^3 (3x^2 - x^3) , dx$. $3x^2 > x^3$ on this interval.
$A = \int_0^3 (3x^2 - x^3) , dx$. $3x^2 > x^3$ on this interval.
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How do you handle regions where the curves intersect multiple times?
How do you handle regions where the curves intersect multiple times?
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Calculate separate integrals for each region. Different regions require separate area calculations.
Calculate separate integrals for each region. Different regions require separate area calculations.
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State the process to find the area between curves crossing multiple times.
State the process to find the area between curves crossing multiple times.
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Divide into segments, integrate each, sum the areas. Handle each interval with proper curve ordering.
Divide into segments, integrate each, sum the areas. Handle each interval with proper curve ordering.
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What is the area between $y = x^2$ and $y = 4x - x^2$ from $x = 0$ to $x = 2$?
What is the area between $y = x^2$ and $y = 4x - x^2$ from $x = 0$ to $x = 2$?
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$A = \int_0^2 [(4x - x^2) - x^2] , dx$. $4x - x^2$ is above $x^2$ on this interval.
$A = \int_0^2 [(4x - x^2) - x^2] , dx$. $4x - x^2$ is above $x^2$ on this interval.
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What is the general formula for the area between two curves $f(x)$ and $g(x)$?
What is the general formula for the area between two curves $f(x)$ and $g(x)$?
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$A = \int_a^b [f(x) - g(x)] , dx$. Subtracts lower curve from upper curve over the interval.
$A = \int_a^b [f(x) - g(x)] , dx$. Subtracts lower curve from upper curve over the interval.
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Identify the upper curve for $y = x^2$ and $y = 3 - x^2$ on $[0, 2]$.
Identify the upper curve for $y = x^2$ and $y = 3 - x^2$ on $[0, 2]$.
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$y = 3 - x^2$ is the upper curve. $3 - x^2$ has larger values on this interval.
$y = 3 - x^2$ is the upper curve. $3 - x^2$ has larger values on this interval.
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How is the area between $y = x^2$ and $y = x^3$ over $[0, 1]$ calculated?
How is the area between $y = x^2$ and $y = x^3$ over $[0, 1]$ calculated?
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$A = \int_0^1 (x^2 - x^3) , dx$. $x^2 > x^3$ for $x \in (0, 1)$.
$A = \int_0^1 (x^2 - x^3) , dx$. $x^2 > x^3$ for $x \in (0, 1)$.
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What integral calculates the area between $y = 4x$ and $y = x^2$ on $[0, 4]$?
What integral calculates the area between $y = 4x$ and $y = x^2$ on $[0, 4]$?
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$A = \int_0^4 (4x - x^2) , dx$. $4x > x^2$ on this interval.
$A = \int_0^4 (4x - x^2) , dx$. $4x > x^2$ on this interval.
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