https://file.aiquickdraw.com/mathbot/file/1729032264388piqf89qg.jpg

The image shows a triangle bounded by a line and the coordinate axes. The task is to find the area of this region using an iterated integral.

### Step 1: Identify the boundary of the region.
The triangle is bounded by:
- The x-axis (from \(x = 0\) to \(x = 3\)).
- The y-axis (from \(y = 0\) to \(y = 9\)).
- The line passing through the points \((0, 9)\) and \((3, 0)\), which forms the hypotenuse of the triangle.

### Step 2: Equation of the line.
We can determine the equation of the line using the two points. The slope is:
\[
m = \frac{0 - 9}{3 - 0} = -3
\]
So the equation of the line is:
\[
y = -3x + 9
\]

### Step 3: Set up the iterated integral.
The limits of integration for \(x\) are from 0 to 3. For each \(x\), the \(y\)-values range from 0 to the line \(y = -3x + 9\).

Thus, the iterated integral to find the area is:
\[
\int_0^3 \int_0^{-3x+9} dy \, dx
\]

### Step 4: Solve the iterated integral.
First, integrate with respect to \(y\):
\[
\int_0^{-3x+9} dy = (-3x + 9)
\]
Now, integrate with respect to \(x\):
\[
\int_0^3 (-3x + 9) dx = \left[\frac{-3x^2}{2} + 9x \right]_0^3
\]
Evaluating at \(x = 3\) and \(x = 0\):
\[
\left( \frac{-3(3)^2}{2} + 9(3) \right) - \left( \frac{-3(0)^2}{2} + 9(0) \right) = \left( \frac{-27}{2} + 27 \right) = \frac{27}{2}
\]

Thus, the area of the region is \( \frac{27}{2} \) or 13.5 square units.

Would you like more details or have any questions?

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Here are 5 related questions to expand on this:
1. How would the limits of integration change if the line's slope were different?
2. Can we confirm the area by geometrically calculating the area of a triangle?
3. What would the iterated integral look like if the region were rotated around the x-axis?
4. How would you solve for the area if the line had curvature instead of being straight?
5. Can the region's area be determined using polar coordinates?

**Tip**: For line equations in slope-intercept form, always use the general form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Use an iterated integral to find the area of the triangular region bounded by the x-axis, y-axis, and the line passing through points (0, 9) and (3, 0).

Find the area of a triangle formed by the x-axis, y-axis, and the line y = -3x + 9 using an iterated integral. Learn step-by-step how to set the limits of integration, solve for the integral, and confirm the area as 13.5 square units.

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