The base of a solid \[S\] is the region bounded by the \[x\]-axis, the \[y\]-axis, and the line through the points \[(0,a)\] and \[(b,0)\].

\[y\]\[x\]\[(0,a)\]\[(b,0)\]

Cross-sections perpendicular to the \[y\]-axis are equilateral triangles.

**Determine the *exact* volume of solid \[S\].**

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.

Learn how to calculate the exact volume of a solid with equilateral triangle cross-sections using mathematical principles. This problem involves integrating the area of equilateral triangles perpendicular to the y-axis. Suitable for advanced high school students and those interested in geometry and calculus.

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