Find the volume of the solid that results from rotating the region bounded by the graphs of y−3x−7=0
, y=0
, and x=2
about the x
-axis. Write the exact answer. Do not round.

To find the volume of the solid obtained by rotating the region bounded by the graphs of \( y - 3x - 7 = 0 \), \( y = 0 \), and \( x = 2 \) about the \( x \)-axis, we will use the **disk method**.

### Step 1: Rearrange the line equation
The equation of the line is:
\[
y - 3x - 7 = 0 \quad \Rightarrow \quad y = 3x + 7.
\]

### Step 2: Set up the bounds
The region is bounded by \( x = 2 \) and the x-axis, which corresponds to \( y = 0 \).

### Step 3: Disk method formula
The formula for the volume \( V \) of a solid of revolution using the disk method is:
\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx,
\]
where \( f(x) \) represents the function being revolved around the x-axis.

Here, \( f(x) = 3x + 7 \), and the limits of integration are from \( x = 0 \) to \( x = 2 \).

### Step 4: Set up the integral
Now, we can write the integral for the volume:
\[
V = \pi \int_0^2 (3x + 7)^2 \, dx.
\]

### Step 5: Expand \( (3x + 7)^2 \)
We expand the square:
\[
(3x + 7)^2 = 9x^2 + 42x + 49.
\]

### Step 6: Integrate
Now, we integrate term by term:
\[
V = \pi \int_0^2 (9x^2 + 42x + 49) \, dx.
\]
This becomes:
\[
V = \pi \left[ 3x^3 + 21x^2 + 49x \right]_0^2.
\]

### Step 7: Evaluate the definite integral
First, evaluate at \( x = 2 \):
\[
3(2)^3 + 21(2)^2 + 49(2) = 3(8) + 21(4) + 49(2) = 24 + 84 + 98 = 206.
\]

Now, evaluate at \( x = 0 \):
\[
3(0)^3 + 21(0)^2 + 49(0) = 0.
\]

### Step 8: Final answer
Thus, the volume is:
\[
V = \pi (206 - 0) = 206\pi.
\]

The exact volume of the solid is \( \boxed{206\pi} \).

Would you like further clarification or additional details on any part of this solution?

Here are some related questions for practice:
1. What is the volume when the same region is rotated around the y-axis?
2. How would the volume change if the line \( y - 3x - 7 = 0 \) was shifted down by 2 units?
3. What is the area of the region bounded by the same curves without rotation?
4. How would you find the volume using the shell method instead?
5. How does the volume formula change if the axis of rotation is the line \( y = -1 \)?

**Tip:** In problems involving rotation, always identify whether to use the disk/washer or shell method depending on the axis of rotation and the region's shape.

Find the volume of the solid that results from rotating the region bounded by the graphs of y - 3x - 7 = 0, y = 0, and x = 2 about the x-axis. Write the exact answer. Do not round.

This detailed solution explains how to find the volume of the solid obtained by rotating the region bounded by the line y = 3x + 7, y = 0, and x = 2 around the x-axis using the disk method. The solution includes step-by-step integration, expansion of binomials, and evaluation of definite integrals. Suitable for advanced calculus students in Grades 11-12.

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