Volumes with Cross Sections: Squares/Rectangles - AP Calculus AB

1. Find the volume for $b(x) = 1$, $h(x) = x$, for $x=0$ to $x=2$ with rectangular cross sections.

   $2$ cubic units. $\int_0^2 1 \cdot x dx = [\frac{x^2}{2}]_0^2 = 2$

2. What is the formula for the volume of a solid with square cross sections?

   Volume = $\text{integral of } (s(x))^2 \text{ dx}$. For square cross sections, area is $s^2$, then integrate over the interval.

3. Find the volume for $s(x) = 3x$ from $x=0$ to $x=1$ with square cross sections.

   $3$ cubic units. $\int_0^1 (3x)^2 dx = \int_0^1 9x^2 dx = 3$

4. Identify $s(x)$ for a square cross section if given the function $y = 3x + 1$.

   Side $s(x) = 3x + 1$. The side length of the square equals the given function.

5. State the relationship between volume and cross-sectional area.

   Volume = $\text{integral of cross-sectional area dx}$. Volume equals the integral of cross-sectional area over the interval.

6. Find the volume with rectangular cross sections $b(x)=2x$, $h(x)=1$, $x=0$ to $x=3$.

   $9$ cubic units. $\int_0^3 2x \cdot 1 dx = [x^2]_0^3 = 9$

7. What does $s(x)$ represent in the formula for square cross sections?

   Side length of the square. $s(x)$ is the length of each side of the square cross section.

8. What is the integral for volume with rectangular cross sections with $b(x)=x$, $h(x)=2$?

   $\text{Integral of } 2x \text{ dx}$. Rectangular area is $b(x) \cdot h(x) = x \cdot 2 = 2x$.

9. What is the integral setup for finding volume with $s(x) = 4$ from $x=0$ to $x=2$?

   $\text{Integral of } 16 \text{ dx from } 0 \text{ to } 2$. $(s(x))^2 = 16$ when $s(x) = 4$, so integrate 16.

10. What is the setup for volume if $s(x) = x+1$ from $x=1$ to $x=4$ with square cross sections?

    $\text{Integral of } (x+1)^2 \text{ dx from 1 to 4}$. Square area is $(s(x))^2 = (x+1)^2$.

11. Find the volume for $s(x) = x+2$ from $x=0$ to $x=3$ with square cross sections.

    $39$ cubic units. $\int_0^3 (x+2)^2 dx = [\frac{(x+2)^3}{3}]_0^3 = \frac{125-8}{3} = 39$

12. What is the integral setup if $b(x) = 2$ and $h(x) = x^2$ from $x=0$ to $x=2$?

    $\text{Integral of } 2x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = 2 \cdot x^2 = 2x^2$.

13. Find the volume for $b(x)=2x$, $h(x)=1$, from $x=0$ to $x=2$ with rectangular cross sections.

    $4$ cubic units. $\int_0^2 2x \cdot 1 dx = [x^2]_0^2 = 4$

14. What is the integral for volume with $s(x)=x^2$ from $x=0$ to $x=1$ with square cross sections?

    $\text{Integral of } x^4 \text{ dx from } 0 \text{ to } 1$. Square area is $ $(s(x))^2$ = $(x^2)^2$ = $x^4$ $

15. Find the volume for $s(x) = 2x+1$ from $x=1$ to $x=3$ with square cross sections.

    $36$ cubic units. $\int_1^3 (2x+1)^2 dx$ evaluates to 36 after expanding and integrating.

16. Find the volume with square cross sections for $s(x) = x$ from $x=0$ to $x=2$.

    $\frac{8}{3}$ cubic units. $\int_0^2 x^2 dx = [\frac{x^3}{3}]_0^2 = \frac{8}{3}$

17. What is the volume for $b(x)=3$, $h(x)=x$, from $x=0$ to $x=1$ with rectangular cross sections?

    $\frac{3}{2}$ cubic units. $\int_0^1 3 \cdot x dx = [\frac{3x^2}{2}]_0^1 = \frac{3}{2}$

18. What is the formula for cross-sectional area of a square?

    Area = $(s(x))^2$. Area of a square is side length squared.

19. What is the meaning of the limit of integration in volume calculations?

    The range over which cross sections are integrated. Integration limits define where cross sections exist along the axis.

20. Which axis do cross sections perpendicular to the x-axis align with?

    The y-axis. Cross sections perpendicular to x-axis are parallel to y-axis.

21. What is the volume if $b(x) = 1$, $h(x) = x^3$, from $x=0$ to $x=1$ with rectangular cross sections?

    $\frac{1}{4}$ cubic units. $\int_0^1 x^3 dx = [\frac{x^4}{4}]_0^1 = \frac{1}{4}$

22. Identify the base function $b(x)$ for a rectangle if given $y = x^2$.

    Base $b(x) = x^2$. The base function is directly given as $y = x^2$.

23. State the setup to find volume if $s(x) = \frac{1}{x}$ from $x=1$ to $x=3$ with square cross sections.

    $\text{Integral of } \frac{1}{x^2} \text{ dx from } 1 \text{ to } 3$. Square area is $(s(x))^2 = (\frac{1}{x})^2 = \frac{1}{x^2}$.

24. Find the volume for $s(x) = \frac{1}{x}$ from $x=1$ to $x=2$ with square cross sections.

    $\frac{1}{2}$ cubic units. $\int_1^2 \frac{1}{x^2} dx = [-\frac{1}{x}]_1^2 = -\frac{1}{2} + 1 = \frac{1}{2}$

25. What is the integral setup for $b(x) = x^2$, $h(x) = 1$, from $x=0$ to $x=2$?

    $\text{Integral of } x^2 \text{ dx from } 0 \text{ to } 2$. Rectangular area is $b(x) \cdot h(x) = x^2 \cdot 1 = x^2$.

26. State the integral for volume if $b(x)=x$, $h(x)=x^3$, from $x=0$ to $x=1$.

    Integral of $x^4$ dx from 0 to 1. Rectangular area is $b(x) \cdot h(x) = x \cdot x^3 = x^4$.

27. What is the integral setup for volume if $b(x) = x$ and $h(x) = x^2$ from $x=0$ to $x=1$?

    $\text{Integral of } x^3 \text{ dx from } 0 \text{ to } 1$. Rectangular area is $b(x) \cdot h(x) = x \cdot x^2 = x^3$.

28. If $b(x) = x$ and $h(x) = 2$, find the volume from $x=1$ to $x=3$.

    $8$ cubic units. $\int_1^3 x \cdot 2 dx = [x^2]_1^3 = 9 - 1 = 8$

29. In what scenario would $s(x)$ be a constant?

    When cross sections are identical squares. All cross sections are identical squares when side length is constant.

30. State the result of integrating $(s(x))^2$ for volume if $s(x) = x^2$ from $x=0$ to $x=1$.

    $\frac{1}{5}$ cubic units. $\int_0^1 x^4 dx = [\frac{x^5}{5}]_0^1 = \frac{1}{5}$

31. What is the meaning of $b(x)$ and $h(x)$ in the rectangular cross section formula?

    $b(x)$ is base, $h(x)$ is height. $b(x)$ is the base length and $h(x)$ is the height of the rectangle.

32. What is the base of the cross section if $b(x) = x^3$?

    Base $b(x) = x^3$. The base function is given directly as $b(x) = x^3$.

33. What is the formula for the volume of a solid with rectangular cross sections?

    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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