Volumes with Cross Sections: Triangles/Semicircles - AP Calculus AB
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Convert the area of a triangle with base $b$ and height $h$ into an integral for volume.
Convert the area of a triangle with base $b$ and height $h$ into an integral for volume.
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Volume = $\frac{1}{2} \times b \times \text{integral of } h \text{ dx}$. Triangle area formula converted to integral form.
Volume = $\frac{1}{2} \times b \times \text{integral of } h \text{ dx}$. Triangle area formula converted to integral form.
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How is the volume of a solid with semicircular cross sections calculated?
How is the volume of a solid with semicircular cross sections calculated?
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Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
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What is the integral for the volume of a solid with triangular cross sections of base $b$?
What is the integral for the volume of a solid with triangular cross sections of base $b$?
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Volume = $\frac{1}{2} \times b \times \text{integral of height dx}$. Base $b$ is constant, height function is integrated.
Volume = $\frac{1}{2} \times b \times \text{integral of height dx}$. Base $b$ is constant, height function is integrated.
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How do you find the volume of a solid with cross sections of varying shape or size?
How do you find the volume of a solid with cross sections of varying shape or size?
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Integrate the area of the cross section along the axis. Integration sums all cross-sectional areas.
Integrate the area of the cross section along the axis. Integration sums all cross-sectional areas.
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What is the general method to find volumes using cross-sections?
What is the general method to find volumes using cross-sections?
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Integrate the area of cross-sections along the axis. Standard approach for volume by cross-sections.
Integrate the area of cross-sections along the axis. Standard approach for volume by cross-sections.
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How is the volume of a solid with triangular cross sections calculated?
How is the volume of a solid with triangular cross sections calculated?
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Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
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What integral represents the volume of a solid with triangular cross sections?
What integral represents the volume of a solid with triangular cross sections?
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Volume = $\frac{1}{2} \times \int \text{Base} \times \text{Height} \text{ dx}$. Triangle cross-sectional area integrated along the axis.
Volume = $\frac{1}{2} \times \int \text{Base} \times \text{Height} \text{ dx}$. Triangle cross-sectional area integrated along the axis.
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How is the volume of a solid with triangular cross sections calculated?
How is the volume of a solid with triangular cross sections calculated?
Tap to reveal answer
Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
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What is the general method to find volumes using cross-sections?
What is the general method to find volumes using cross-sections?
Tap to reveal answer
Integrate the area of cross-sections along the axis. Standard approach for volume by cross-sections.
Integrate the area of cross-sections along the axis. Standard approach for volume by cross-sections.
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How do you find the volume of a solid with cross sections of varying shape or size?
How do you find the volume of a solid with cross sections of varying shape or size?
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Integrate the area of the cross section along the axis. Integration sums all cross-sectional areas.
Integrate the area of the cross section along the axis. Integration sums all cross-sectional areas.
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Convert the area of a triangle with base $b$ and height $h$ into an integral for volume.
Convert the area of a triangle with base $b$ and height $h$ into an integral for volume.
Tap to reveal answer
Volume = $\frac{1}{2} \times b \times \text{integral of } h \text{ dx}$. Triangle area formula converted to integral form.
Volume = $\frac{1}{2} \times b \times \text{integral of } h \text{ dx}$. Triangle area formula converted to integral form.
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What integral represents the volume of a solid with triangular cross sections?
What integral represents the volume of a solid with triangular cross sections?
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Volume = $\frac{1}{2} \times \text{integral of } \text{Base} \times \text{Height} \text{ dx}$. Triangle cross-sectional area integrated along the axis.
Volume = $\frac{1}{2} \times \text{integral of } \text{Base} \times \text{Height} \text{ dx}$. Triangle cross-sectional area integrated along the axis.
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What is the integral for the volume of a solid with triangular cross sections of base $b$?
What is the integral for the volume of a solid with triangular cross sections of base $b$?
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Volume = $\frac{1}{2} \times b \times \text{integral of height dx}$. Base $b$ is constant, height function is integrated.
Volume = $\frac{1}{2} \times b \times \text{integral of height dx}$. Base $b$ is constant, height function is integrated.
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How is the volume of a solid with semicircular cross sections calculated?
How is the volume of a solid with semicircular cross sections calculated?
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Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.
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What is $A(x)$ for a semicircle if the radius is $r(x)=x+1$?
What is $A(x)$ for a semicircle if the radius is $r(x)=x+1$?
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$A(x)=\frac{\pi}{2}(x+1)^2$. Substitutes $r=x+1$ into semicircle area formula $A=\frac{\pi}{2}r^2$.
$A(x)=\frac{\pi}{2}(x+1)^2$. Substitutes $r=x+1$ into semicircle area formula $A=\frac{\pi}{2}r^2$.
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What is $A(x)$ for a right isosceles triangle if the leg is $s(x)=3-x$?
What is $A(x)$ for a right isosceles triangle if the leg is $s(x)=3-x$?
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$A(x)=\frac{1}{2}(3-x)^2$. Substitutes $s=3-x$ into right isosceles triangle area formula.
$A(x)=\frac{1}{2}(3-x)^2$. Substitutes $s=3-x$ into right isosceles triangle area formula.
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What is $A(x)$ for an equilateral triangle if the side length is $s(x)=\sqrt{x}$?
What is $A(x)$ for an equilateral triangle if the side length is $s(x)=\sqrt{x}$?
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$A(x)=\frac{\sqrt{3}}{4}x$. Substitutes $s=\sqrt{x}$ into $A=\frac{\sqrt{3}}{4}s^2$ to get $\frac{\sqrt{3}}{4}x$.
$A(x)=\frac{\sqrt{3}}{4}x$. Substitutes $s=\sqrt{x}$ into $A=\frac{\sqrt{3}}{4}s^2$ to get $\frac{\sqrt{3}}{4}x$.
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What are the correct bounds for $x$ if the base region is between $y=x$ and $y=x^2$ with vertical slices?
What are the correct bounds for $x$ if the base region is between $y=x$ and $y=x^2$ with vertical slices?
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$x\in[0,1]$. Curves intersect where $x=x^2$, so $x=0$ and $x=1$.
$x\in[0,1]$. Curves intersect where $x=x^2$, so $x=0$ and $x=1$.
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Identify the correct setup for semicircle cross sections: base between $y=f(x)$ and $y=g(x)$ on $[a,b]$.
Identify the correct setup for semicircle cross sections: base between $y=f(x)$ and $y=g(x)$ on $[a,b]$.
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$V=\int_a^b \frac{\pi}{8}(f(x)-g(x))^2,dx$. Combines volume integral with semicircle area using diameter as base.
$V=\int_a^b \frac{\pi}{8}(f(x)-g(x))^2,dx$. Combines volume integral with semicircle area using diameter as base.
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Find the diameter $d(x)$ if the base region is between $y=\sqrt{x}$ (top) and $y=x$ (bottom).
Find the diameter $d(x)$ if the base region is between $y=\sqrt{x}$ (top) and $y=x$ (bottom).
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$d(x)=\sqrt{x}-x$. Top curve minus bottom curve gives vertical distance.
$d(x)=\sqrt{x}-x$. Top curve minus bottom curve gives vertical distance.
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What is the area of a semicircle cross section with radius $r$ (in terms of $r$)?
What is the area of a semicircle cross section with radius $r$ (in terms of $r$)?
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$A=\frac{\pi}{2}r^2$. Half the area of a full circle with radius $r$.
$A=\frac{\pi}{2}r^2$. Half the area of a full circle with radius $r$.
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What is the area of a right isosceles triangular cross section with leg length $s$?
What is the area of a right isosceles triangular cross section with leg length $s$?
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$A=\frac{1}{2}s^2$. For isosceles right triangle, area is half the square of the leg.
$A=\frac{1}{2}s^2$. For isosceles right triangle, area is half the square of the leg.
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What is the area of an equilateral triangular cross section with side length $s$?
What is the area of an equilateral triangular cross section with side length $s$?
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$A=\frac{\sqrt{3}}{4}s^2$. Standard formula for equilateral triangle area using side length.
$A=\frac{\sqrt{3}}{4}s^2$. Standard formula for equilateral triangle area using side length.
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Which variable do you integrate with if cross sections are perpendicular to the $x$-axis?
Which variable do you integrate with if cross sections are perpendicular to the $x$-axis?
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$dx$. Cross sections perpendicular to $x$-axis vary with $x$.
$dx$. Cross sections perpendicular to $x$-axis vary with $x$.
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Which variable do you integrate with if cross sections are perpendicular to the $y$-axis?
Which variable do you integrate with if cross sections are perpendicular to the $y$-axis?
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$dy$. Cross sections perpendicular to $y$-axis vary with $y$.
$dy$. Cross sections perpendicular to $y$-axis vary with $y$.
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What is the diameter function $d(x)$ if the base is the vertical distance between $y=f(x)$ and $y=g(x)$?
What is the diameter function $d(x)$ if the base is the vertical distance between $y=f(x)$ and $y=g(x)$?
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$d(x)=f(x)-g(x)$. Vertical distance between curves gives diameter at each $x$.
$d(x)=f(x)-g(x)$. Vertical distance between curves gives diameter at each $x$.
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What is the side/leg length $s(x)$ if a cross section uses the base segment between $y=f(x)$ and $y=g(x)$?
What is the side/leg length $s(x)$ if a cross section uses the base segment between $y=f(x)$ and $y=g(x)$?
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$s(x)=f(x)-g(x)$. Vertical distance between curves gives side/leg length at each $x$.
$s(x)=f(x)-g(x)$. Vertical distance between curves gives side/leg length at each $x$.
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Identify the area function $A(x)$ for semicircular cross sections with diameter $d(x)$.
Identify the area function $A(x)$ for semicircular cross sections with diameter $d(x)$.
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$A(x)=\frac{\pi}{8}(d(x))^2$. Substitutes diameter function into semicircle area formula.
$A(x)=\frac{\pi}{8}(d(x))^2$. Substitutes diameter function into semicircle area formula.
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Identify the area function $A(x)$ for right isosceles triangles with leg $s(x)$ as the base segment.
Identify the area function $A(x)$ for right isosceles triangles with leg $s(x)$ as the base segment.
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$A(x)=\frac{1}{2}(s(x))^2$. Substitutes leg function into right isosceles triangle area formula.
$A(x)=\frac{1}{2}(s(x))^2$. Substitutes leg function into right isosceles triangle area formula.
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Identify the area function $A(x)$ for equilateral triangles with side $s(x)$ as the base segment.
Identify the area function $A(x)$ for equilateral triangles with side $s(x)$ as the base segment.
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$A(x)=\frac{\sqrt{3}}{4}(s(x))^2$. Substitutes side function into equilateral triangle area formula.
$A(x)=\frac{\sqrt{3}}{4}(s(x))^2$. Substitutes side function into equilateral triangle area formula.
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