Area Between Curves: Functions of x - AP Calculus AB
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Calculate the area between $y = x + 1$ and $y = 2x$ from $x = 0$ to $x = 1$.
Calculate the area between $y = x + 1$ and $y = 2x$ from $x = 0$ to $x = 1$.
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$\frac{1}{2}$. Same calculation as previous duplicate problem.
$\frac{1}{2}$. Same calculation as previous duplicate problem.
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State the condition for $f(x)$ and $g(x)$ to find area between curves.
State the condition for $f(x)$ and $g(x)$ to find area between curves.
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$f(x) \geq g(x)$ over interval $[a, b]$. Upper function must be greater than or equal to lower function.
$f(x) \geq g(x)$ over interval $[a, b]$. Upper function must be greater than or equal to lower function.
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Find the intersection points of $y = x^2$ and $y = 4$.
Find the intersection points of $y = x^2$ and $y = 4$.
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$x = -2, x = 2$. Solve $x^2 = 4$ to get $x = \pm 2$.
$x = -2, x = 2$. Solve $x^2 = 4$ to get $x = \pm 2$.
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Identify the upper function for $f(x) = x^2 + 1$ and $g(x) = x^2$ on $[0, 1]$.
Identify the upper function for $f(x) = x^2 + 1$ and $g(x) = x^2$ on $[0, 1]$.
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$f(x) = x^2 + 1$. Shifted parabola is above original parabola.
$f(x) = x^2 + 1$. Shifted parabola is above original parabola.
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What is the integral for the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$?
What is the integral for the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$?
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$\int_{0}^{1} [x - x^2] , dx$. Line is above parabola on unit interval.
$\int_{0}^{1} [x - x^2] , dx$. Line is above parabola on unit interval.
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Calculate the area between $y = 4$ and $y = x^2$ from $x = -2$ to $x = 2$.
Calculate the area between $y = 4$ and $y = x^2$ from $x = -2$ to $x = 2$.
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$\frac{32}{3}$. Evaluate $\int_{-2}^2 (4-x^2)dx$ using symmetry.
$\frac{32}{3}$. Evaluate $\int_{-2}^2 (4-x^2)dx$ using symmetry.
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Which function is lower: $y = x^3$ or $y = x^2$ on $[0, 1]$?
Which function is lower: $y = x^3$ or $y = x^2$ on $[0, 1]$?
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$y = x^3$. Cubic grows slower than quadratic on $[0,1]$.
$y = x^3$. Cubic grows slower than quadratic on $[0,1]$.
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What is the integral for the area between $y = x^2$ and $y = -x$ from $x = 0$ to $x = 1$?
What is the integral for the area between $y = x^2$ and $y = -x$ from $x = 0$ to $x = 1$?
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$\int_{0}^{1} [x^2 - (-x)] , dx$. Parabola is above line $y = -x$ on this interval.
$\int_{0}^{1} [x^2 - (-x)] , dx$. Parabola is above line $y = -x$ on this interval.
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Identify the lower function for $f(x) = x^2 + 1$ and $g(x) = x + 2$ on $[0, 1]$.
Identify the lower function for $f(x) = x^2 + 1$ and $g(x) = x + 2$ on $[0, 1]$.
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$f(x) = x^2 + 1$. Shifted parabola is below line on this interval.
$f(x) = x^2 + 1$. Shifted parabola is below line on this interval.
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Determine the integral for the area between $y = x^2$ and $y = 4$ from $x = -2$ to $x = 2$.
Determine the integral for the area between $y = x^2$ and $y = 4$ from $x = -2$ to $x = 2$.
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$\int_{-2}^{2} [4 - x^2] , dx$. Horizontal line is above parabola from $-2$ to $2$.
$\int_{-2}^{2} [4 - x^2] , dx$. Horizontal line is above parabola from $-2$ to $2$.
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What is the integral to find the area between $y = x^2$ and $y = 4$ from $x = 0$ to $x = 2$?
What is the integral to find the area between $y = x^2$ and $y = 4$ from $x = 0$ to $x = 2$?
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$\int_{0}^{2} [4 - x^2] , dx$. Horizontal line is above parabola on this interval.
$\int_{0}^{2} [4 - x^2] , dx$. Horizontal line is above parabola on this interval.
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How do you find the limits of integration for two curves $f(x)$ and $g(x)$?
How do you find the limits of integration for two curves $f(x)$ and $g(x)$?
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Solve $f(x) = g(x)$. Set functions equal and solve for intersection points.
Solve $f(x) = g(x)$. Set functions equal and solve for intersection points.
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Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$.
Find the area between $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$.
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$\frac{1}{6}$. Evaluate $\int_0^1 (x-x^2)dx$ for line above parabola.
$\frac{1}{6}$. Evaluate $\int_0^1 (x-x^2)dx$ for line above parabola.
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How do you express the area between $y = 3x$ and $y = x^2$ on $[0, 3]$?
How do you express the area between $y = 3x$ and $y = x^2$ on $[0, 3]$?
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$\int_{0}^{3} [3x - x^2] , dx$. Line $y=3x$ is above parabola on $[0,3]$.
$\int_{0}^{3} [3x - x^2] , dx$. Line $y=3x$ is above parabola on $[0,3]$.
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Calculate the area between $y = x^2 - 1$ and $y = 2x - 1$ from $x = 0$ to $x = 1$.
Calculate the area between $y = x^2 - 1$ and $y = 2x - 1$ from $x = 0$ to $x = 1$.
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$\frac{1}{6}$. Line $y=2x-1$ is above shifted parabola on $[0,1]$.
$\frac{1}{6}$. Line $y=2x-1$ is above shifted parabola on $[0,1]$.
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What is the geometric interpretation of $\int_{a}^{b} [g(x) - f(x)] , dx$?
What is the geometric interpretation of $\int_{a}^{b} [g(x) - f(x)] , dx$?
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Area between $g(x)$ and $f(x)$ on $[a, b]$. When $g(x) \geq f(x)$, integral gives area between curves.
Area between $g(x)$ and $f(x)$ on $[a, b]$. When $g(x) \geq f(x)$, integral gives area between curves.
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What does $\int_{a}^{b} [f(x) - g(x)] , dx$ represent geometrically?
What does $\int_{a}^{b} [f(x) - g(x)] , dx$ represent geometrically?
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Area between $f(x)$ and $g(x)$ on $[a, b]$. The definite integral represents signed area between curves.
Area between $f(x)$ and $g(x)$ on $[a, b]$. The definite integral represents signed area between curves.
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Identify the lower function for $f(x) = x^2$ and $g(x) = x + 2$ on $[2, 3]$.
Identify the lower function for $f(x) = x^2$ and $g(x) = x + 2$ on $[2, 3]$.
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$f(x) = x^2$. Parabola is below linear function on this interval.
$f(x) = x^2$. Parabola is below linear function on this interval.
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Identify the upper function for $f(x) = x^3$ and $g(x) = x$ on $[-1, 1]$.
Identify the upper function for $f(x) = x^3$ and $g(x) = x$ on $[-1, 1]$.
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$g(x) = x$. Line is above cubic on symmetric interval around origin.
$g(x) = x$. Line is above cubic on symmetric interval around origin.
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What is the area formula for two curves $f(x)$ and $g(x)$ over $[a, b]$?
What is the area formula for two curves $f(x)$ and $g(x)$ over $[a, b]$?
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$\int_{a}^{b} |f(x) - g(x)| , dx$. Absolute value ensures positive area regardless of function order.
$\int_{a}^{b} |f(x) - g(x)| , dx$. Absolute value ensures positive area regardless of function order.
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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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$\frac{8}{3}$. Same calculation as previous identical problem.
$\frac{8}{3}$. Same calculation as previous identical problem.
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Which function is lower: $y = x^3$ or $y = x$ on $[-1, 0]$?
Which function is lower: $y = x^3$ or $y = x$ on $[-1, 0]$?
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$y = x^3$. Cubic function is negative while line is positive on $[-1,0]$.
$y = x^3$. Cubic function is negative while line is positive on $[-1,0]$.
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Calculate the area between $y = x^2 + 1$ and $y = 1$ from $x = -1$ to $x = 1$.
Calculate the area between $y = x^2 + 1$ and $y = 1$ from $x = -1$ to $x = 1$.
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$\frac{8}{3}$. Same calculation as before for shifted parabola.
$\frac{8}{3}$. Same calculation as before for shifted parabola.
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What is the integral setup for the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$?
What is the integral setup for the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$?
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$\int_{0}^{3} [3x - x^2] , dx$. Same setup as previous identical problem.
$\int_{0}^{3} [3x - x^2] , dx$. Same setup as previous identical problem.
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Identify the upper function for $f(x) = 2x$ and $g(x) = x^2$ on $[0, 2]$.
Identify the upper function for $f(x) = 2x$ and $g(x) = x^2$ on $[0, 2]$.
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$f(x) = 2x$. Line has greater slope than parabola on $[0,2]$.
$f(x) = 2x$. Line has greater slope than parabola on $[0,2]$.
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What is the integral to find the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$?
What is the integral to find the area between $y = 3x$ and $y = x^2$ from $x = 0$ to $x = 3$?
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$\int_{0}^{3} [3x - x^2] , dx$. Line is above parabola on the given interval.
$\int_{0}^{3} [3x - x^2] , dx$. Line is above parabola on the given interval.
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Calculate the area between $y = 2x$ and $y = x^2$ from $x = 0$ to $x = 2$.
Calculate the area between $y = 2x$ and $y = x^2$ from $x = 0$ to $x = 2$.
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$\frac{8}{3}$. Line $y=2x$ is above parabola on $[0,2]$.
$\frac{8}{3}$. Line $y=2x$ is above parabola on $[0,2]$.
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What is the purpose of finding intersection points of $f(x)$ and $g(x)$?
What is the purpose of finding intersection points of $f(x)$ and $g(x)$?
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To determine limits of integration. Intersection points become the integration bounds.
To determine limits of integration. Intersection points become the integration bounds.
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Determine the area between $y = x^3$ and $y = x$ from $x = 0$ to $x = 1$.
Determine the area between $y = x^3$ and $y = x$ from $x = 0$ to $x = 1$.
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$\frac{1}{4}$. Line $y=x$ is above cubic $y=x^3$ on $[0,1]$.
$\frac{1}{4}$. Line $y=x$ is above cubic $y=x^3$ on $[0,1]$.
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What is the first step to find the area between two curves $f(x)$ and $g(x)$?
What is the first step to find the area between two curves $f(x)$ and $g(x)$?
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Identify intersection points of $f(x)$ and $g(x)$. Intersection points determine the limits of integration.
Identify intersection points of $f(x)$ and $g(x)$. Intersection points determine the limits of integration.
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