The problem asks to find the volume of a solid whose base is the region bounded by the curves $y = x^2$ and $y = \sqrt{x}$, with cross-sections perpendicular to the x-axis being squares.

Steps to solve the problem:

1. Understand the Region: The curves given are $y = x^2$ (a parabola) and $y = \sqrt{x}$ (a square root function). The region of interest is where these two curves intersect.

   To find the intersection points: $x^2 = \sqrt{x}$ Squaring both sides to eliminate the square root: $x^4 = x$ Rearranging this equation: $x^4 - x = 0$ Factor the equation: $x(x^3 - 1) = 0$ This gives $x = 0$ or $x^3 = 1$, so $x = 1$.

   Thus, the region is bounded between $x = 0$ and $x = 1$.

2. Cross-Sectional Area: The cross-sections perpendicular to the x-axis are squares. The side length of each square is the vertical distance between the two curves at a given $x$, which is: $\text{Side length} = \sqrt{x} - x^2$ Since the area of a square is $\text{side length}^2$, the area of the cross-section at $x$ is: $A(x) = \left( \sqrt{x} - x^2 \right)^2$

3. Volume Calculation: To find the volume of the solid, integrate the area of the cross-sections from $x = 0$ to $x = 1$: $V = \int_0^1 \left( \sqrt{x} - x^2 \right)^2 dx$

4. Simplify the Integrand: Expand $\left( \sqrt{x} - x^2 \right)^2$: $\left( \sqrt{x} - x^2 \right)^2 = x - 2x^{5/2} + x^4$

5. Integrate: Now, integrate each term separately: $V = \int_0^1 \left( x - 2x^{5/2} + x^4 \right) dx$

   • $\int_0^1 x \, dx = \frac{1}{2}$
   • $\int_0^1 x^{5/2} \, dx = \frac{2}{7}$
   • $\int_0^1 x^4 \, dx = \frac{1}{5}$

   Therefore: $V = \frac{1}{2} - 2 \times \frac{2}{7} + \frac{1}{5}$ Simplify: $V = \frac{1}{2} - \frac{4}{7} + \frac{1}{5}$ Finding a common denominator (70): $V = \frac{35}{70} - \frac{40}{70} + \frac{14}{70}$ $V = \frac{35 - 40 + 14}{70} = \frac{9}{70}$

Thus, the volume of the solid is $\frac{9}{70}$.

Would you like further details or clarifications on any of these steps?

Related Questions:

1. What is the general formula for calculating volumes using cross-sectional areas?
2. How do we handle cross-sections of different shapes, such as equilateral triangles or semicircles?
3. How does the region of intersection between curves affect the volume of a solid?
4. What happens to the volume if the cross-sections are parallel to the y-axis instead of the x-axis?
5. How would the calculation change if the cross-sections were rectangles instead of squares?

Tip:

Always check the bounds of integration carefully, especially when curves intersect, to ensure you're integrating over the correct region.