Volumes with Cross Sections: Squares/Rectangles - AP Calculus AB
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What is the base of the cross section if $b(x) = x^3$?
What is the base of the cross section if $b(x) = x^3$?
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Base $b(x) = x^3$. The base function is given directly as $b(x) = x^3$.
Base $b(x) = x^3$. The base function is given directly as $b(x) = x^3$.
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Find the volume with rectangular cross sections $b(x)=2x$, $h(x)=1$, $x=0$ to $x=3$.
Find the volume with rectangular cross sections $b(x)=2x$, $h(x)=1$, $x=0$ to $x=3$.
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$9$ cubic units. $\int_0^3 2x \cdot 1 dx = [x^2]_0^3 = 9$
$9$ cubic units. $\int_0^3 2x \cdot 1 dx = [x^2]_0^3 = 9$
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Find the volume for $b(x)=2x$, $h(x)=1$, from $x=0$ to $x=2$ with rectangular cross sections.
Find the volume for $b(x)=2x$, $h(x)=1$, from $x=0$ to $x=2$ with rectangular cross sections.
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$4$ cubic units. $\int_0^2 2x \cdot 1 dx = [x^2]_0^2 = 4$
$4$ cubic units. $\int_0^2 2x \cdot 1 dx = [x^2]_0^2 = 4$
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What is the integral setup for $b(x) = x^2$, $h(x) = 1$, from $x=0$ to $x=2$?
What is the integral setup for $b(x) = x^2$, $h(x) = 1$, from $x=0$ to $x=2$?
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$\text{Integral of } x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = x^2 \cdot 1 = x^2$.
$\text{Integral of } x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = x^2 \cdot 1 = x^2$.
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State the integral for volume if $b(x)=x$, $h(x)=x^3$, from $x=0$ to $x=1$.
State the integral for volume if $b(x)=x$, $h(x)=x^3$, from $x=0$ to $x=1$.
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Integral of $x^4$ dx from 0 to 1. Rectangular area is $b(x) \cdot h(x) = x \cdot x^3 = x^4$.
Integral of $x^4$ dx from 0 to 1. Rectangular area is $b(x) \cdot h(x) = x \cdot x^3 = x^4$.
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Find the volume for $s(x) = 2x+1$ from $x=1$ to $x=3$ with square cross sections.
Find the volume for $s(x) = 2x+1$ from $x=1$ to $x=3$ with square cross sections.
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$36$ cubic units. $\int_1^3 (2x+1)^2 dx$ evaluates to 36 after expanding and integrating.
$36$ cubic units. $\int_1^3 (2x+1)^2 dx$ evaluates to 36 after expanding and integrating.
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State the result of integrating $(s(x))^2$ for volume if $s(x) = x^2$ from $x=0$ to $x=1$.
State the result of integrating $(s(x))^2$ for volume if $s(x) = x^2$ from $x=0$ to $x=1$.
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$\frac{1}{5}$ cubic units. $\int_0^1 x^4 dx = [\frac{x^5}{5}]_0^1 = \frac{1}{5}$
$\frac{1}{5}$ cubic units. $\int_0^1 x^4 dx = [\frac{x^5}{5}]_0^1 = \frac{1}{5}$
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In what scenario would $s(x)$ be a constant?
In what scenario would $s(x)$ be a constant?
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When cross sections are identical squares. All cross sections are identical squares when side length is constant.
When cross sections are identical squares. All cross sections are identical squares when side length is constant.
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Find the volume with square cross sections for $s(x) = x$ from $x=0$ to $x=2$.
Find the volume with square cross sections for $s(x) = x$ from $x=0$ to $x=2$.
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$\frac{8}{3}$ cubic units. $\int_0^2 x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{8}{3}$
$\frac{8}{3}$ cubic units. $\int_0^2 x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{8}{3}$
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Find the volume for $s(x) = 3x$ from $x=0$ to $x=1$ with square cross sections.
Find the volume for $s(x) = 3x$ from $x=0$ to $x=1$ with square cross sections.
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$3$ cubic units. $\int_0^1 (3x)^2 dx = \int_0^1 9x^2 dx = 3$
$3$ cubic units. $\int_0^1 (3x)^2 dx = \int_0^1 9x^2 dx = 3$
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What is the volume if $b(x) = 1$, $h(x) = x^3$, from $x=0$ to $x=1$ with rectangular cross sections?
What is the volume if $b(x) = 1$, $h(x) = x^3$, from $x=0$ to $x=1$ with rectangular cross sections?
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$\frac{1}{4}$ cubic units. $\int_0^1 x^3 dx = [\frac{x^4}{4}]_0^1 = \frac{1}{4}$
$\frac{1}{4}$ cubic units. $\int_0^1 x^3 dx = [\frac{x^4}{4}]_0^1 = \frac{1}{4}$
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What is the volume for $b(x)=3$, $h(x)=x$, from $x=0$ to $x=1$ with rectangular cross sections?
What is the volume for $b(x)=3$, $h(x)=x$, from $x=0$ to $x=1$ with rectangular cross sections?
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$\frac{3}{2}$ cubic units. $\int_0^1 3 \cdot x dx = [\frac{3x^2}{2}]_0^1 = \frac{3}{2}$
$\frac{3}{2}$ cubic units. $\int_0^1 3 \cdot x dx = [\frac{3x^2}{2}]_0^1 = \frac{3}{2}$
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Find the volume for $s(x) = x+2$ from $x=0$ to $x=3$ with square cross sections.
Find the volume for $s(x) = x+2$ from $x=0$ to $x=3$ with square cross sections.
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$39$ cubic units. $\int_0^3 (x+2)^2 dx = [\frac{(x+2)^3}{3}]_0^3 = \frac{125-8}{3} = 39$
$39$ cubic units. $\int_0^3 (x+2)^2 dx = [\frac{(x+2)^3}{3}]_0^3 = \frac{125-8}{3} = 39$
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What is the formula for cross-sectional area of a square?
What is the formula for cross-sectional area of a square?
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Area = $(s(x))^2$. Area of a square is side length squared.
Area = $(s(x))^2$. Area of a square is side length squared.
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Find the volume for $b(x) = 1$, $h(x) = x$, for $x=0$ to $x=2$ with rectangular cross sections.
Find the volume for $b(x) = 1$, $h(x) = x$, for $x=0$ to $x=2$ with rectangular cross sections.
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$2$ cubic units. $\int_0^2 1 \cdot x dx = [\frac{x^2}{2}]_0^2 = 2$
$2$ cubic units. $\int_0^2 1 \cdot x dx = [\frac{x^2}{2}]_0^2 = 2$
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What is the integral setup if $b(x) = 2$ and $h(x) = x^2$ from $x=0$ to $x=2$?
What is the integral setup if $b(x) = 2$ and $h(x) = x^2$ from $x=0$ to $x=2$?
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$\text{Integral of } 2x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = 2 \cdot x^2 = 2x^2$.
$\text{Integral of } 2x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = 2 \cdot x^2 = 2x^2$.
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What is the meaning of $b(x)$ and $h(x)$ in the rectangular cross section formula?
What is the meaning of $b(x)$ and $h(x)$ in the rectangular cross section formula?
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$b(x)$ is base, $h(x)$ is height. $b(x)$ is the base length and $h(x)$ is the height of the rectangle.
$b(x)$ is base, $h(x)$ is height. $b(x)$ is the base length and $h(x)$ is the height of the rectangle.
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What is the integral setup for finding volume with $s(x) = 4$ from $x=0$ to $x=2$?
What is the integral setup for finding volume with $s(x) = 4$ from $x=0$ to $x=2$?
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$\text{Integral of } 16 \text{ dx from } 0 \text{ to } 2$. $(s(x))^2 = 16$ when $s(x) = 4$, so integrate 16.
$\text{Integral of } 16 \text{ dx from } 0 \text{ to } 2$. $(s(x))^2 = 16$ when $s(x) = 4$, so integrate 16.
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Find the volume for $s(x) = \frac{1}{x}$ from $x=1$ to $x=2$ with square cross sections.
Find the volume for $s(x) = \frac{1}{x}$ from $x=1$ to $x=2$ with square cross sections.
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$\frac{1}{2}$ cubic units. $\int_1^2 \frac{1}{x^2} dx = [-\frac{1}{x}]_1^2 = -\frac{1}{2} + 1 = \frac{1}{2}$
$\frac{1}{2}$ cubic units. $\int_1^2 \frac{1}{x^2} dx = [-\frac{1}{x}]_1^2 = -\frac{1}{2} + 1 = \frac{1}{2}$
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State the setup to find volume if $s(x) = \frac{1}{x}$ from $x=1$ to $x=3$ with square cross sections.
State the setup to find volume if $s(x) = \frac{1}{x}$ from $x=1$ to $x=3$ with square cross sections.
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$\text{Integral of } \frac{1}{x^2} \text{ dx from } 1 \text{ to } 3$. Square area is $(s(x))^2 = (\frac{1}{x})^2 = \frac{1}{x^2}$.
$\text{Integral of } \frac{1}{x^2} \text{ dx from } 1 \text{ to } 3$. Square area is $(s(x))^2 = (\frac{1}{x})^2 = \frac{1}{x^2}$.
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Which axis do cross sections perpendicular to the x-axis align with?
Which axis do cross sections perpendicular to the x-axis align with?
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The y-axis. Cross sections perpendicular to x-axis are parallel to y-axis.
The y-axis. Cross sections perpendicular to x-axis are parallel to y-axis.
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What is the meaning of the limit of integration in volume calculations?
What is the meaning of the limit of integration in volume calculations?
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The range over which cross sections are integrated. Integration limits define where cross sections exist along the axis.
The range over which cross sections are integrated. Integration limits define where cross sections exist along the axis.
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Identify the base function $b(x)$ for a rectangle if given $y = x^2$.
Identify the base function $b(x)$ for a rectangle if given $y = x^2$.
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Base $b(x) = x^2$. The base function is directly given as $y = x^2$.
Base $b(x) = x^2$. The base function is directly given as $y = x^2$.
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What does $s(x)$ represent in the formula for square cross sections?
What does $s(x)$ represent in the formula for square cross sections?
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Side length of the square. $s(x)$ is the length of each side of the square cross section.
Side length of the square. $s(x)$ is the length of each side of the square cross section.
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What is the integral for volume with rectangular cross sections with $b(x)=x$, $h(x)=2$?
What is the integral for volume with rectangular cross sections with $b(x)=x$, $h(x)=2$?
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$\text{Integral of } 2x \text{ dx}$. Rectangular area is $b(x) \cdot h(x) = x \cdot 2 = 2x$.
$\text{Integral of } 2x \text{ dx}$. Rectangular area is $b(x) \cdot h(x) = x \cdot 2 = 2x$.
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If $b(x) = x$ and $h(x) = 2$, find the volume from $x=1$ to $x=3$.
If $b(x) = x$ and $h(x) = 2$, find the volume from $x=1$ to $x=3$.
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$8$ cubic units. $\int_1^3 x \cdot 2 dx = [x^2]_1^3 = 9 - 1 = 8$
$8$ cubic units. $\int_1^3 x \cdot 2 dx = [x^2]_1^3 = 9 - 1 = 8$
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What is the formula for the volume of a solid with rectangular cross sections?
What is the formula for the volume of a solid with rectangular cross sections?
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Volume $= \text{integral of } (b(x) \times h(x)) \text{ dx}$. For rectangles, area is base times height, then integrate over the interval.
Volume $= \text{integral of } (b(x) \times h(x)) \text{ dx}$. For rectangles, area is base times height, then integrate over the interval.
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What is the integral setup for volume if $b(x) = x$ and $h(x) = x^2$ from $x=0$ to $x=1$?
What is the integral setup for volume if $b(x) = x$ and $h(x) = x^2$ from $x=0$ to $x=1$?
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$\text{Integral of } x^3 \text{ dx from } 0 \text{ to } 1$. Rectangular area is $b(x) \cdot h(x) = x \cdot x^2 = x^3$.
$\text{Integral of } x^3 \text{ dx from } 0 \text{ to } 1$. Rectangular area is $b(x) \cdot h(x) = x \cdot x^2 = x^3$.
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Identify $s(x)$ for a square cross section if given the function $y = 3x + 1$.
Identify $s(x)$ for a square cross section if given the function $y = 3x + 1$.
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Side $s(x) = 3x + 1$. The side length of the square equals the given function.
Side $s(x) = 3x + 1$. The side length of the square equals the given function.
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What is the integral for volume with $s(x)=x^2$ from $x=0$ to $x=1$ with square cross sections?
What is the integral for volume with $s(x)=x^2$ from $x=0$ to $x=1$ with square cross sections?
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$\text{Integral of } x^4 \text{ dx from } 0 \text{ to } 1$. Square area is $ (s(x))^2 = (x^2)^2 = x^4 $
$\text{Integral of } x^4 \text{ dx from } 0 \text{ to } 1$. Square area is $ (s(x))^2 = (x^2)^2 = x^4 $
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