To find the volume of the solid whose base is bounded by the curves $4x = y^2$, $5x = y^2$, and $x = 4$ in the first quadrant, and whose cross-sections perpendicular to the x-axis are squares, we can follow these steps:

Step 1: Identify the bounds of the base region

The given curves are:

• $4x = y^2$ or $y = \sqrt{4x}$,
• $5x = y^2$ or $y = \sqrt{5x}$,
• The line $x = 4$.

These curves and the line form the boundary of the base of the solid in the first quadrant.

Step 2: Determine the length of the side of the square cross-sections

Since the cross-sections perpendicular to the x-axis are squares, the length of the side of each square at a given $x$ will be the vertical distance between the two curves. The two curves are $y = \sqrt{5x}$ (the larger curve) and $y = \sqrt{4x}$ (the smaller curve).

Thus, the side length $s(x)$ of the square at a point $x$ is: $s(x) = \sqrt{5x} - \sqrt{4x}.$

Step 3: Set up the volume integral

The volume $V$ of the solid can be found by integrating the area of the cross-sections along the x-axis from $x = 0$ to $x = 4$. The area of each square cross-section is $s(x)^2$, so the volume is: $V = \int_0^4 \left[ \sqrt{5x} - \sqrt{4x} \right]^2 \, dx.$

Step 4: Simplify the integrand

First, simplify the expression for $s(x)^2$: $s(x) = \sqrt{5x} - \sqrt{4x},$ $s(x)^2 = (\sqrt{5x} - \sqrt{4x})^2 = (5x + 4x - 2\sqrt{5x} \cdot \sqrt{4x}) = 9x - 2 \cdot 2 \sqrt{20x} = 9x - 4\sqrt{20x}.$

Thus, the volume integral becomes: $V = \int_0^4 \left( 9x - 4\sqrt{20x} \right) \, dx.$

Step 5: Compute the integral

We now compute each part of the integral:

1. $\int_0^4 9x \, dx = 9 \left[ \frac{x^2}{2} \right]_0^4 = 9 \cdot \frac{16}{2} = 72$.
2. For the second term, $\int_0^4 4\sqrt{20x} \, dx$, we can factor out constants: [ 4\sqrt{20} \int_0^4 \sqrt{x} , dx = 4\sqrt{20} \cdot \left[ \frac{2}{3} x^{3/2} \right]_0^4 = 4\sqrt{20} \cdot \frac{2}{3} \left[ 4^{3/2} - 0 \right]. ] Since $4^{3/2} = 8$, this becomes: $4\sqrt{20} \cdot \frac{2}{3} \cdot 8 = 4\sqrt{20} \cdot \frac{16}{3} = \frac{64 \sqrt{20}}{3}.$ Simplifying $\sqrt{20} = 2\sqrt{5}$, we get: $\frac{64 \cdot 2\sqrt{5}}{3} = \frac{128 \sqrt{5}}{3}.$

Step 6: Final answer

Thus, the total volume is: $V = 72 - \frac{128 \sqrt{5}}{3}.$

This is the volume of the solid.

Would you like a further breakdown of the steps or clarification on any part of the solution? Also, here are 5 related questions:

1. How do you calculate the area of a square given its side length?
2. What is the general approach for finding the volume of a solid with square cross-sections?
3. How do you integrate functions involving square roots?
4. How would the volume change if the cross-sections were semicircles instead of squares?
5. What happens to the volume if the curves were shifted along the x-axis or y-axis?

Tip: When working with integrals of square root functions, look for opportunities to simplify the expression before integrating.