The problem consists of two parts. Let's tackle each part in detail.

Part 1

The volume of a solid whose base is bounded by $y = \sqrt{x}$ and $y = x^2$, with vertical cross-sections perpendicular to the $x$-axis shaped as semicircles.

1. Determine the bounds of $x$: Find the intersection points of $y = \sqrt{x}$ and $y = x^2$ by setting $\sqrt{x} = x^2$. $x = x^4 \Rightarrow x^4 - x = 0 \Rightarrow x(x^3 - 1) = 0$ This gives $x = 0$ and $x = 1$.

2. Area of cross-section: For a given $x$ between 0 and 1, the length of the diameter of the semicircle is given by the distance between $y = \sqrt{x}$ and $y = x^2$: $\text{Diameter} = \sqrt{x} - x^2$ The radius $r$ of the semicircle is: $r = \frac{\sqrt{x} - x^2}{2}$ The area $A(x)$ of the semicircular cross-section is: $A(x) = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \left(\frac{\sqrt{x} - x^2}{2}\right)^2 = \frac{\pi}{8} (\sqrt{x} - x^2)^2$

3. Volume of the solid: The volume $V$ is the integral of the cross-sectional area from $x = 0$ to $x = 1$: $V = \int_0^1 A(x) \, dx = \int_0^1 \frac{\pi}{8} (\sqrt{x} - x^2)^2 \, dx$ Expanding $(\sqrt{x} - x^2)^2$ gives: $(\sqrt{x} - x^2)^2 = x - 2x^{5/2} + x^4$ Thus, $V = \frac{\pi}{8} \int_0^1 (x - 2x^{5/2} + x^4) \, dx$ Integrate each term: $V = \frac{\pi}{8} \left[ \frac{x^2}{2} - \frac{2x^{7/2}}{7} + \frac{x^5}{5} \right]_0^1$ $V = \frac{\pi}{8} \left( \frac{1}{2} - \frac{2}{7} + \frac{1}{5} \right)$ Simplify this to find the exact volume.

Part 2

The volume of the intersection of two quarter-cylinders of radius 1.

The shape created is a three-dimensional region where two quarter-cylinders intersect at right angles. This shape is known as the Steinmetz solid.

The volume $V$ of the intersection of two quarter-cylinders with radius $r = 1$ can be found using the formula: $V = 2 \times \left( \frac{1}{4} \pi r^2 \right)^2$ For $r = 1$: $V = 2 \times \left( \frac{1}{4} \pi \cdot 1^2 \right)^2$ Compute this to find the volume.

Would you like a step-by-step calculation for each part or help with any specific step?

Questions to Explore Further

1. How does the intersection of functions affect the volume in a solid of revolution?
2. What are the applications of integrals in determining volumes of solids with different cross-sectional shapes?
3. How can different shapes of cross-sections (e.g., squares, triangles) affect the overall volume calculation?
4. What other methods exist for calculating the volume of the Steinmetz solid?
5. How does symmetry simplify the computation of intersection volumes in geometry?

Tip

Using symmetry in volume integrals can greatly simplify calculations, especially with intersecting solids.