To find the volume of the solid $S$ with the given conditions, we follow these steps:

Step 1: Determine the equation of the line

The base of the solid $S$ is the region bounded by the $x$-axis, the $y$-axis, and the line passing through the points $(0, a)$ and $(b, 0)$. The equation of this line can be found using the slope-intercept form:

$\text{Slope} = \frac{0 - a}{b - 0} = -\frac{a}{b}$ So, the equation of the line is: $y = -\frac{a}{b}x + a$ or equivalently, $x = \frac{b}{a}(a - y)$

Step 2: Cross-sectional area of the equilateral triangle

Cross-sections perpendicular to the $y$-axis are equilateral triangles. The side length $s(y)$ of each triangle is the length of the segment from the line to the $y$-axis, which corresponds to the length $x$ given by: $s(y) = \frac{b}{a}(a - y)$ The area $A(y)$ of an equilateral triangle with side length $s(y)$ is given by: $A(y) = \frac{\sqrt{3}}{4} s(y)^2$ Substituting the expression for $s(y)$: $A(y) = \frac{\sqrt{3}}{4} \left(\frac{b}{a}(a - y)\right)^2$ Simplify the expression: $A(y) = \frac{\sqrt{3}}{4} \cdot \frac{b^2}{a^2} (a - y)^2$

Step 3: Set up the integral for the volume

The volume $V$ of the solid is found by integrating the cross-sectional area along the $y$-axis, from $y = 0$ to $y = a$: $V = \int_{0}^{a} A(y) \, dy = \int_{0}^{a} \frac{\sqrt{3}}{4} \cdot \frac{b^2}{a^2} (a - y)^2 \, dy$ Factor out the constants: $V = \frac{\sqrt{3}}{4} \cdot \frac{b^2}{a^2} \int_{0}^{a} (a - y)^2 \, dy$

Step 4: Evaluate the integral

The integral $\int_{0}^{a} (a - y)^2 \, dy$ can be computed using a substitution. Let $u = a - y$, so $du = -dy$. The limits change as follows: when $y = 0$, $u = a$; when $y = a$, $u = 0$.

Thus, the integral becomes: $\int_{0}^{a} (a - y)^2 \, dy = -\int_{a}^{0} u^2 \, du = \int_{0}^{a} u^2 \, du$ Now, integrate $u^2$: $\int u^2 \, du = \frac{u^3}{3}$ Evaluate from 0 to $a$: $\left[\frac{u^3}{3}\right]_{0}^{a} = \frac{a^3}{3}$ So, the integral is: $\int_{0}^{a} (a - y)^2 \, dy = \frac{a^3}{3}$

Step 5: Find the volume

Substitute this result into the expression for $V$: $V = \frac{\sqrt{3}}{4} \cdot \frac{b^2}{a^2} \cdot \frac{a^3}{3}$ Simplify: $V = \frac{\sqrt{3}}{12} \cdot b^2 \cdot a$

Final Answer

The exact volume of the solid $S$ is: $V = \frac{\sqrt{3}}{12} a b^2$

Would you like further details on any step or have any questions?

Here are some related questions:

1. How does the area formula for an equilateral triangle derive from the side length?
2. Can you solve the integral $\int (a - y)^2 \, dy$ using a different method?
3. How would the volume change if the cross-sections were squares instead of triangles?
4. What is the significance of the constants $\sqrt{3}/4$ and $1/12$ in the context of equilateral triangles?
5. How does the method of slicing help in calculating volumes of other solids?

Tip: When dealing with volumes by slicing, always confirm the orientation of the cross-sections and the corresponding limits of integration.