Volumes with Cross Sections: Triangles/Semicircles - AP Calculus BC
Card 1 of 30
Find the volume of a solid with triangular cross sections, base $= 4$, height $= 5$, length $= 6$.
Find the volume of a solid with triangular cross sections, base $= 4$, height $= 5$, length $= 6$.
Tap to reveal answer
Volume = 60. Apply volume formula: $\frac{1}{2} \times 4 \times 5 \times 6$.
Volume = 60. Apply volume formula: $\frac{1}{2} \times 4 \times 5 \times 6$.
← Didn't Know|Knew It →
What is the volume of a solid with semicircular cross sections where the diameter is 10 and length is 3?
What is the volume of a solid with semicircular cross sections where the diameter is 10 and length is 3?
Tap to reveal answer
Volume = $37.5\text{π}$. Radius 5, area $\frac{25\pi}{2}$, times length 3.
Volume = $37.5\text{π}$. Radius 5, area $\frac{25\pi}{2}$, times length 3.
← Didn't Know|Knew It →
If $f(x) = x^2$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=3$?
If $f(x) = x^2$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=3$?
Tap to reveal answer
Radius = 4.5. Half the distance from $3^2=9$ to 0 gives $9/2=4.5$.
Radius = 4.5. Half the distance from $3^2=9$ to 0 gives $9/2=4.5$.
← Didn't Know|Knew It →
What is the role of the base function $f(x)$ in determining cross sections?
What is the role of the base function $f(x)$ in determining cross sections?
Tap to reveal answer
$f(x)$ provides the upper boundary. Defines the top edge of each cross-section.
$f(x)$ provides the upper boundary. Defines the top edge of each cross-section.
← Didn't Know|Knew It →
If $f(x) = x^2$ and $g(x) = 0$, what is the base of the triangular cross section at $x=2$?
If $f(x) = x^2$ and $g(x) = 0$, what is the base of the triangular cross section at $x=2$?
Tap to reveal answer
Base = 4. Distance from $x^2$ to 0 at $x=2$ gives $4-0=4$.
Base = 4. Distance from $x^2$ to 0 at $x=2$ gives $4-0=4$.
← Didn't Know|Knew It →
Find the volume of a solid with semicircular cross sections, radius $= 4$, and length $= 5$.
Find the volume of a solid with semicircular cross sections, radius $= 4$, and length $= 5$.
Tap to reveal answer
Volume = $40\text{π}$. Semicircle area $\frac{\pi \times 4^2}{2}=8\pi$ times length 5.
Volume = $40\text{π}$. Semicircle area $\frac{\pi \times 4^2}{2}=8\pi$ times length 5.
← Didn't Know|Knew It →
What is the volume of a solid with triangular cross sections where base and height are both constant at 5?
What is the volume of a solid with triangular cross sections where base and height are both constant at 5?
Tap to reveal answer
Volume = $12.5 \times \text{Length}$. Triangle area $\frac{1}{2} \times 5 \times 5 = 12.5$ times length.
Volume = $12.5 \times \text{Length}$. Triangle area $\frac{1}{2} \times 5 \times 5 = 12.5$ times length.
← Didn't Know|Knew It →
Find the volume of a solid with triangular cross sections, base $= 9$, height $= 12$, length $= 10$.
Find the volume of a solid with triangular cross sections, base $= 9$, height $= 12$, length $= 10$.
Tap to reveal answer
Volume = 540. Apply formula: $\frac{1}{2} \times 9 \times 12 \times 10$.
Volume = 540. Apply formula: $\frac{1}{2} \times 9 \times 12 \times 10$.
← Didn't Know|Knew It →
If $f(x) = x^3$ and $g(x) = x$, find the base of the triangular cross section at $x=1$.
If $f(x) = x^3$ and $g(x) = x$, find the base of the triangular cross section at $x=1$.
Tap to reveal answer
Base = 0. At $x=1$: $1^3-1=0$, so base is 0.
Base = 0. At $x=1$: $1^3-1=0$, so base is 0.
← Didn't Know|Knew It →
Compute the volume for a solid with semicircular cross sections with radius $r = 2$ and length $l = 6$.
Compute the volume for a solid with semicircular cross sections with radius $r = 2$ and length $l = 6$.
Tap to reveal answer
Volume = $12\text{π}$. Semicircle area $\frac{\pi \times 2^2}{2}=2\pi$ times length 6.
Volume = $12\text{π}$. Semicircle area $\frac{\pi \times 2^2}{2}=2\pi$ times length 6.
← Didn't Know|Knew It →
If $f(x) = x^3$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=1$?
If $f(x) = x^3$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=1$?
Tap to reveal answer
Radius = 0.5. At $x=1$: $\frac{1^3-0}{2}=\frac{1}{2}=0.5$.
Radius = 0.5. At $x=1$: $\frac{1^3-0}{2}=\frac{1}{2}=0.5$.
← Didn't Know|Knew It →
Compute the volume of a solid with triangular cross sections, base $= 6$, height $= 4$, length $= 8$.
Compute the volume of a solid with triangular cross sections, base $= 6$, height $= 4$, length $= 8$.
Tap to reveal answer
Volume = 96. Apply formula: $\frac{1}{2} \times 6 \times 4 \times 8$.
Volume = 96. Apply formula: $\frac{1}{2} \times 6 \times 4 \times 8$.
← Didn't Know|Knew It →
Calculate the volume of a solid with semicircular cross sections, diameter $= 6$, length $= 4$.
Calculate the volume of a solid with semicircular cross sections, diameter $= 6$, length $= 4$.
Tap to reveal answer
Volume = $18\text{π}$. Using semicircle formula with radius 3 and length 4.
Volume = $18\text{π}$. Using semicircle formula with radius 3 and length 4.
← Didn't Know|Knew It →
Given $f(x) = e^x$ and $g(x) = 0$, find the height of the triangular cross section at $x=0$.
Given $f(x) = e^x$ and $g(x) = 0$, find the height of the triangular cross section at $x=0$.
Tap to reveal answer
Height = 1. At $x=0$: $e^0-0=1$, giving height 1.
Height = 1. At $x=0$: $e^0-0=1$, giving height 1.
← Didn't Know|Knew It →
What is the formula for the radius of a semicircle given diameter $d$?
What is the formula for the radius of a semicircle given diameter $d$?
Tap to reveal answer
Radius = $\frac{d}{2}$. Basic relationship between radius and diameter.
Radius = $\frac{d}{2}$. Basic relationship between radius and diameter.
← Didn't Know|Knew It →
What is the volume of a solid with semicircular cross sections where the radius is constant at 3?
What is the volume of a solid with semicircular cross sections where the radius is constant at 3?
Tap to reveal answer
Volume = $9\text{π} \times \text{Length}$. Semicircle area $\frac{\pi r^2}{2}$ with $r=3$ times length.
Volume = $9\text{π} \times \text{Length}$. Semicircle area $\frac{\pi r^2}{2}$ with $r=3$ times length.
← Didn't Know|Knew It →
Given $f(x) = x+1$ and $g(x) = x$, find the height of the triangular cross section at $x=1$.
Given $f(x) = x+1$ and $g(x) = x$, find the height of the triangular cross section at $x=1$.
Tap to reveal answer
Height = 1. Distance between functions: $(1+1)-(1)=1$.
Height = 1. Distance between functions: $(1+1)-(1)=1$.
← Didn't Know|Knew It →
Given $f(x) = x^2 + 1$ and $g(x) = x$, find the diameter of the semicircular cross section at $x=2$.
Given $f(x) = x^2 + 1$ and $g(x) = x$, find the diameter of the semicircular cross section at $x=2$.
Tap to reveal answer
Diameter = 3. Distance between $(2^2+1)=5$ and $2$ gives diameter 3.
Diameter = 3. Distance between $(2^2+1)=5$ and $2$ gives diameter 3.
← Didn't Know|Knew It →
What is the role of the base function $g(x)$ in determining cross sections?
What is the role of the base function $g(x)$ in determining cross sections?
Tap to reveal answer
$g(x)$ provides the lower boundary. Defines the bottom edge of each cross-section.
$g(x)$ provides the lower boundary. Defines the bottom edge of each cross-section.
← Didn't Know|Knew It →
Find the volume of a solid with semicircular cross sections, diameter $= 8$ and length $= 2$.
Find the volume of a solid with semicircular cross sections, diameter $= 8$ and length $= 2$.
Tap to reveal answer
Volume = $16\text{π}$. Radius 4, area $8\pi$, times length 2.
Volume = $16\text{π}$. Radius 4, area $8\pi$, times length 2.
← Didn't Know|Knew It →
What is the formula for the area of a semicircle with diameter $d$?
What is the formula for the area of a semicircle with diameter $d$?
Tap to reveal answer
Area = $\frac{\text{π} \times (d/2)^2}{2}$. Semicircle area using diameter instead of radius.
Area = $\frac{\text{π} \times (d/2)^2}{2}$. Semicircle area using diameter instead of radius.
← Didn't Know|Knew It →
Calculate the volume of a solid with semicircular cross sections, radius $= 3$, and length $= 7$.
Calculate the volume of a solid with semicircular cross sections, radius $= 3$, and length $= 7$.
Tap to reveal answer
Volume = $31.5\text{π}$. Semicircle area $\frac{9\pi}{2}$ times length 7.
Volume = $31.5\text{π}$. Semicircle area $\frac{9\pi}{2}$ times length 7.
← Didn't Know|Knew It →
What is the volume of a solid with triangular cross sections, base $= 7$ and height $= 8$?
What is the volume of a solid with triangular cross sections, base $= 7$ and height $= 8$?
Tap to reveal answer
Volume = $28 \times \text{Length}$. Triangle area $\frac{1}{2} \times 7 \times 8 = 28$ times length.
Volume = $28 \times \text{Length}$. Triangle area $\frac{1}{2} \times 7 \times 8 = 28$ times length.
← Didn't Know|Knew It →
What is the volume formula for a solid with equilateral triangular cross sections?
What is the volume formula for a solid with equilateral triangular cross sections?
Tap to reveal answer
Volume = $\frac{\text{√3}}{4} \times \text{side}^2 \times \text{Length}$. Uses equilateral triangle area formula with side length.
Volume = $\frac{\text{√3}}{4} \times \text{side}^2 \times \text{Length}$. Uses equilateral triangle area formula with side length.
← Didn't Know|Knew It →
Identify the radius for semicircular cross sections in the plane $x = a$.
Identify the radius for semicircular cross sections in the plane $x = a$.
Tap to reveal answer
Radius = $\frac{f(a) - g(a)}{2}$. Half the distance between boundary functions.
Radius = $\frac{f(a) - g(a)}{2}$. Half the distance between boundary functions.
← Didn't Know|Knew It →
What is the diameter for semicircular cross sections if base function $f(x) = x + 2$ and $g(x) = x$?
What is the diameter for semicircular cross sections if base function $f(x) = x + 2$ and $g(x) = x$?
Tap to reveal answer
Diameter = 2. Distance between functions $(x+2)-x=2$.
Diameter = 2. Distance between functions $(x+2)-x=2$.
← Didn't Know|Knew It →
Given the function $y = \text{sin}(x)$, find the base of triangle cross section at $x=\frac{\text{π}}{2}$.
Given the function $y = \text{sin}(x)$, find the base of triangle cross section at $x=\frac{\text{π}}{2}$.
Tap to reveal answer
Base = 1. At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2})=1$ gives base of 1.
Base = 1. At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2})=1$ gives base of 1.
← Didn't Know|Knew It →
Identify the base for triangular cross sections in the plane $x = a$.
Identify the base for triangular cross sections in the plane $x = a$.
Tap to reveal answer
Base = $f(a) - g(a)$. Distance between upper and lower boundary functions.
Base = $f(a) - g(a)$. Distance between upper and lower boundary functions.
← Didn't Know|Knew It →
Compute the volume of a solid with triangular cross sections, base $= 6$, height $= 4$, length $= 8$.
Compute the volume of a solid with triangular cross sections, base $= 6$, height $= 4$, length $= 8$.
Tap to reveal answer
Volume = 96. Apply formula: $\frac{1}{2} \times 6 \times 4 \times 8$.
Volume = 96. Apply formula: $\frac{1}{2} \times 6 \times 4 \times 8$.
← Didn't Know|Knew It →
Given the function $y = \text{sin}(x)$, find the base of triangle cross section at $x=\frac{\text{π}}{2}$.
Given the function $y = \text{sin}(x)$, find the base of triangle cross section at $x=\frac{\text{π}}{2}$.
Tap to reveal answer
Base = 1. At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2})=1$ gives base of 1.
Base = 1. At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2})=1$ gives base of 1.
← Didn't Know|Knew It →