To find the volume of the given solid, we need to use the method of cross-sections. Let's break it down step by step:

Step 1: Understand the Region

The base of the solid is bounded by the curve $y = 5 - 5x^2$ and the line $y = 0$, which is the x-axis.

This region is a parabola opening downward with its vertex at $(0, 5)$ and crossing the x-axis at $x = \pm 1$ (since $5 - 5x^2 = 0$ gives $x = \pm 1$).

Step 2: Cross-sectional Area

The cross sections of the solid are described as right triangles, with the base of the triangle lying along the region defined by the curve and the x-axis. Importantly, the height and the base of each right triangle are equal.

At any point $x$, the base of the right triangle is the distance from the curve to the x-axis, which is $y = 5 - 5x^2$.

Thus, the base of the triangle is: $b(x) = 5 - 5x^2$ Since the height equals the base, the height is also: $h(x) = 5 - 5x^2$

The area $A(x)$ of the triangle is given by the formula for the area of a triangle: $A(x) = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting $b(x)$ and $h(x)$ into this: $A(x) = \frac{1}{2} \times (5 - 5x^2) \times (5 - 5x^2) = \frac{1}{2} \times (5 - 5x^2)^2$

Step 3: Set up the Integral

To find the volume, we integrate the cross-sectional area along the x-axis from $x = -1$ to $x = 1$: $V = \int_{-1}^{1} A(x) \, dx = \int_{-1}^{1} \frac{1}{2} (5 - 5x^2)^2 \, dx$

Step 4: Solve the Integral

First, expand $(5 - 5x^2)^2$: $(5 - 5x^2)^2 = 25 - 50x^2 + 25x^4$

Now, substitute this into the integral: $V = \frac{1}{2} \int_{-1}^{1} (25 - 50x^2 + 25x^4) \, dx$

Since the integrand is an even function, we can simplify the limits and multiply by 2: $V = \int_{0}^{1} (25 - 50x^2 + 25x^4) \, dx$

Now integrate term by term: $V = \left[ 25x - \frac{50x^3}{3} + \frac{25x^5}{5} \right]_0^1$

Evaluating at $x = 1$ and $x = 0$: $V = \left( 25(1) - \frac{50(1)^3}{3} + \frac{25(1)^5}{5} \right) - \left( 25(0) - \frac{50(0)^3}{3} + \frac{25(0)^5}{5} \right)$ $V = 25 - \frac{50}{3} + 5 = 30 - \frac{50}{3}$

Simplifying the result: $V = \frac{90}{3} - \frac{50}{3} = \frac{40}{3}$

Final Answer:

The volume of the solid is: $\boxed{\frac{40}{3}}$

Would you like to explore the method of cross-sections further or ask any questions?

Related Questions:

1. How is the method of cross-sections applied to solids with different cross-sectional shapes?
2. What is the difference between using disks, washers, and cross-sections to find volume?
3. Can we apply this technique to solids with circular cross-sections?
4. How does the symmetry of the region affect the computation of the volume?
5. How do you find the volume of a solid when the cross sections are equilateral triangles?

Tip:

Always check the symmetry of the region to potentially simplify your volume calculations by adjusting the limits of integration.