Volumes with Cross Sections: Triangles/Semicircles - AP Calculus BC

1. Find the volume of a solid with triangular cross sections, base = 4, height = 5, length = 6.

   Volume = 60. Apply volume formula: $\frac{1}{2} \times 4 \times 5 \times 6$.

2. What is the diameter for semicircular cross sections if base function $f(x) = x + 2$ and $g(x) = x$?

   Diameter = 2. Distance between functions $(x+2)-x=2$.

3. Given $f(x) = e^x$ and $g(x) = 0$, find the height of the triangular cross section at $x=0$.

   Height = 1. At $x=0$: $e^0-0=1$, giving height 1.

4. Compute the volume of a solid with triangular cross sections, base $= 6$, height $= 4$, length $= 8$.

   Volume = 96. Apply formula: $\frac{1}{2} \times 6 \times 4 \times 8$.

5. Identify the base for triangular cross sections in the plane $x = a$.

   Base = $f(a) - g(a)$. Distance between upper and lower boundary functions.

6. If $f(x) = x^3$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=1$?

   Radius = 0.5. At $x=1$: $\frac{1^3-0}{2}=\frac{1}{2}=0.5$.

7. Calculate the volume of a solid with semicircular cross sections, radius $= 3$, and length $= 7$.

   Volume = $31.5\text{π}$. Semicircle area $\frac{9\pi}{2}$ times length 7.

8. Find the volume of a solid with semicircular cross sections, diameter $= 8$ and length $= 2$.

   Volume = $16\text{π}$. Radius 4, area $8\pi$, times length 2.

9. What is the formula for the radius of a semicircle given diameter $d$?

   Radius = $\frac{d}{2}$. Basic relationship between radius and diameter.

10. What is the area of a triangle with base $b$ and height $h$?

    Area = $\frac{1}{2} \times b \times h$. Standard triangle area formula.

11. Find the volume of a solid with triangular cross sections, base $= 9$, height $= 12$, length $= 10$.

    Volume = 540. Apply formula: $\frac{1}{2} \times 9 \times 12 \times 10$.

12. What is the formula for the area of a semicircle with diameter $d$?

    Area = $\frac{\text{π} \times (d/2)^2}{2}$. Semicircle area using diameter instead of radius.

13. What is the volume of a solid with triangular cross sections where base and height are both constant at 5?

    Volume = $12.5 \times \text{Length}$. Triangle area $\frac{1}{2} \times 5 \times 5 = 12.5$ times length.

14. If $f(x) = x^2$ and $g(x) = 0$, what is the radius of the semicircular cross section at $x=3$?

    Radius = 4.5. Half the distance from $3^2=9$ to 0 gives $9/2=4.5$.

15. Given $f(x) = x^2 + 1$ and $g(x) = x$, find the diameter of the semicircular cross section at $x=2$.

    Diameter = 3. Distance between $(2^2+1)=5$ and $2$ gives diameter 3.

16. Given the function $y = \text{sin}(x)$, find the base of triangle cross section at $x=\frac{\text{π}}{2}$.

    Base = 1. At $x=\frac{\pi}{2}$, $\sin(\frac{\pi}{2})=1$ gives base of 1.

17. If $f(x) = x^3$ and $g(x) = x$, find the base of the triangular cross section at $x=1$.

    Base = 0. At $x=1$: $1^3-1=0$, so base is 0.

18. Find the volume of a solid with semicircular cross sections, radius $= 4$, and length $= 5$.

    Volume = $40\text{π}$. Semicircle area $\frac{\pi \times 4^2}{2}=8\pi$ times length 5.

19. What is the volume of a solid with semicircular cross sections where the radius is constant at 3?

    Volume = $9\text{π} \times \text{Length}$. Semicircle area $\frac{\pi r^2}{2}$ with $r=3$ times length.

20. What is the role of the base function $g(x)$ in determining cross sections?

    $g(x)$ provides the lower boundary. Defines the bottom edge of each cross-section.

21. Identify the radius for semicircular cross sections in the plane $x = a$.

    Radius = $\frac{f(a) - g(a)}{2}$. Half the distance between boundary functions.

22. Compute the volume for a solid with semicircular cross sections with radius $r = 2$ and length $l = 6$.

    Volume = $12\text{π}$. Semicircle area $\frac{\pi \times 2^2}{2}=2\pi$ times length 6.

23. What is the volume of a solid with semicircular cross sections where the diameter is 10 and length is 3?

    Volume = $37.5\text{π}$. Radius 5, area $\frac{25\pi}{2}$, times length 3.

24. If $f(x) = x^2$ and $g(x) = 0$, what is the base of the triangular cross section at $x=2$?

    Base = 4. Distance from $x^2$ to 0 at $x=2$ gives $4-0=4$.

25. What is the volume of a solid with triangular cross sections, base $= 7$ and height $= 8$?

    Volume = $28 \times \text{Length}$. Triangle area $\frac{1}{2} \times 7 \times 8 = 28$ times length.

26. Given $f(x) = x+1$ and $g(x) = x$, find the height of the triangular cross section at $x=1$.

    Height = 1. Distance between functions: $(1+1)-(1)=1$.

27. Find the volume of a solid with triangular cross sections, base $= 4$, height $= 5$, length $= 6$.

    Volume = 60. Apply volume formula: $\frac{1}{2} \times 4 \times 5 \times 6$.

28. What is the role of the base function $f(x)$ in determining cross sections?

    $f(x)$ provides the upper boundary. Defines the top edge of each cross-section.

29. What is the volume formula for a solid with equilateral triangular cross sections?

    Volume = $\frac{\text{√3}}{4} \times \text{side}^2 \times \text{Length}$. Uses equilateral triangle area formula with side length.

30. Calculate the volume of a solid with semicircular cross sections, diameter $= 6$, length $= 4$.

    Volume = $18\text{π}$. Using semicircle formula with radius 3 and length 4.

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