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

1. What is the side/leg length $s(x)$ if a cross section uses the base segment between $y=f(x)$ and $y=g(x)$?

   $s(x)=f(x)-g(x)$. Vertical distance between curves gives side/leg length at each $x$.

2. What is $A(x)$ for a semicircle if the radius is $r(x)=x+1$?

   $A(x)=\frac{\pi}{2}(x+1)^2$. Substitutes $r=x+1$ into semicircle area formula $A=\frac{\pi}{2}r^2$.

3. How is the volume of a solid with semicircular cross sections calculated?

   Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.

4. What is the general method to find volumes using cross-sections?

   Integrate the area of cross-sections along the axis. Standard approach for volume by cross-sections.

5. How is the volume of a solid with triangular cross sections calculated?

   Integrate the area function of the cross sections along the axis. Volume equals integral of cross-sectional areas.

6. Which variable do you integrate with if cross sections are perpendicular to the $y$-axis?

   $dy$. Cross sections perpendicular to $y$-axis vary with $y$.

7. Find the diameter $d(x)$ if the base region is between $y=\sqrt{x}$ (top) and $y=x$ (bottom).

   $d(x)=\sqrt{x}-x$. Top curve minus bottom curve gives vertical distance.

8. What is the area of a triangle cross section when base $b$ and height $h$ are given?

   $A=\frac{1}{2}bh$. Standard triangle area formula: half base times height.

9. Identify the correct setup for semicircle cross sections: base between $y=f(x)$ and $y=g(x)$ on $[a,b]$.

   $V=\int_a^b \frac{\pi}{8}(f(x)-g(x))^2\,dx$. Combines volume integral with semicircle area using diameter as base.

10. What is the diameter function $d(x)$ if the base is the vertical distance between $y=f(x)$ and $y=g(x)$?

    $d(x)=f(x)-g(x)$. Vertical distance between curves gives diameter at each $x$.

11. What is the integral for the volume of a solid with triangular cross sections of base $b$?

    Volume = $\frac{1}{2} \times b \times \text{integral of height dx}$. Base $b$ is constant, height function is integrated.

12. Identify the area function $A(x)$ for right isosceles triangles with leg $s(x)$ as the base segment.

    $A(x)=\frac{1}{2}(s(x))^2$. Substitutes leg function into right isosceles triangle area formula.

13. What is the area of a semicircle cross section with radius $r$ (in terms of $r$)?

    $A=\frac{\pi}{2}r^2$. Half the area of a full circle with radius $r$.

14. What is $A(x)$ for a semicircle if the diameter is $d(x)=2x$?

    $A(x)=\frac{\pi}{2}x^2$. Substitutes $d=2x$ into $A=\frac{\pi}{8}d^2$ to get $\frac{\pi}{8}(2x)^2$.

15. Convert the area of a triangle with base $b$ and height $h$ into an integral for volume.

    Volume = $\frac{1}{2} \times b \times \text{integral of } h \text{ dx}$. Triangle area formula converted to integral form.

16. Identify the area function $A(x)$ for equilateral triangles with side $s(x)$ as the base segment.

    $A(x)=\frac{\sqrt{3}}{4}(s(x))^2$. Substitutes side function into equilateral triangle area formula.

17. Identify the area function $A(x)$ for semicircular cross sections with diameter $d(x)$.

    $A(x)=\frac{\pi}{8}(d(x))^2$. Substitutes diameter function into semicircle area formula.

18. What is the area of an equilateral triangular cross section with side length $s$?

    $A=\frac{\sqrt{3}}{4}s^2$. Standard formula for equilateral triangle area using side length.

19. What is the area of a semicircle cross section with diameter $d$ (in terms of $d$)?

    $A=\frac{\pi}{8}d^2$. Half of a circle's area $\pi r^2$ where $r=\frac{d}{2}$.

20. What are the correct bounds for $x$ if the base region is between $y=x$ and $y=x^2$ with vertical slices?

    $x\in[0,1]$. Curves intersect where $x=x^2$, so $x=0$ and $x=1$.

21. What is $A(x)$ for an equilateral triangle if the side length is $s(x)=\sqrt{x}$?

    $A(x)=\frac{\sqrt{3}}{4}x$. Substitutes $s=\sqrt{x}$ into $A=\frac{\sqrt{3}}{4}s^2$ to get $\frac{\sqrt{3}}{4}x$.

22. What is the general volume integral for known cross-sectional area $A(x)$ on $[a,b]$?

    $V=\int_a^b A(x)\,dx$. Integrates cross-sectional area along the axis to find total volume.

23. What is the area of a right isosceles triangular cross section with leg length $s$?

    $A=\frac{1}{2}s^2$. For isosceles right triangle, area is half the square of the leg.

24. What integral represents the volume of a solid with triangular cross sections?

    Volume = $\frac{1}{2} \times \int \text{Base} \times \text{Height} \text{ dx}$. Triangle cross-sectional area integrated along the axis.

25. Which variable do you integrate with if cross sections are perpendicular to the $x$-axis?

    $dx$. Cross sections perpendicular to $x$-axis vary with $x$.

26. What is $A(x)$ for a right isosceles triangle if the leg is $s(x)=3-x$?

    $A(x)=\frac{1}{2}(3-x)^2$. Substitutes $s=3-x$ into right isosceles triangle area formula.

27. How do you find the volume of a solid with cross sections of varying shape or size?

    Integrate the area of the cross section along the axis. Integration sums all cross-sectional areas.

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