Find the volume of the solid obtained by rotating the region bounded by the given curves about the *x*-axis.

*y* = 8 − 8*x*2, *y* = 0

To find the volume of the solid obtained by rotating the region bounded by the curves \( y = 8 - 8x^2 \) and \( y = 0 \) about the \( x \)-axis, we use the **disk method**. 

The volume \( V \) of the solid formed by revolving a curve \( y = f(x) \) around the \( x \)-axis from \( x = a \) to \( x = b \) is given by the formula:

\[
V = \pi \int_{a}^{b} [f(x)]^2 \, dx
\]

Here, \( f(x) = 8 - 8x^2 \) and \( y = 0 \) is the boundary, so the region we're rotating lies between these two. Now let's go through the steps:

### Step 1: Find the bounds \( a \) and \( b \)
We need to find the \( x \)-values where the curve intersects the \( x \)-axis. This happens when \( y = 0 \), so:

\[
8 - 8x^2 = 0
\]
\[
8x^2 = 8 \implies x^2 = 1 \implies x = \pm 1
\]

Thus, the limits of integration are \( x = -1 \) and \( x = 1 \).

### Step 2: Set up the integral
Using the formula for the disk method, we square \( f(x) \) and set up the integral:

\[
V = \pi \int_{-1}^{1} (8 - 8x^2)^2 \, dx
\]

### Step 3: Solve the integral

First, expand the square:

\[
(8 - 8x^2)^2 = 64 - 128x^2 + 64x^4
\]

Now, integrate term by term:

\[
V = \pi \int_{-1}^{1} \left( 64 - 128x^2 + 64x^4 \right) dx
\]

### Step 4: Compute the integrals

Since the function is even (symmetric around the \( y \)-axis), we can double the integral from 0 to 1 to simplify the calculation:

\[
V = 2\pi \int_{0}^{1} \left( 64 - 128x^2 + 64x^4 \right) dx
\]

Now, calculate each integral:

\[
\int 64 \, dx = 64x
\]
\[
\int -128x^2 \, dx = -\frac{128x^3}{3}
\]
\[
\int 64x^4 \, dx = \frac{64x^5}{5}
\]

Evaluating these from 0 to 1:

\[
64(1) - \frac{128(1^3)}{3} + \frac{64(1^5)}{5} = 64 - \frac{128}{3} + \frac{64}{5}
\]

Now, simplify this expression:

\[
64 - \frac{128}{3} + \frac{64}{5} = \frac{960}{15} - \frac{640}{15} + \frac{192}{15} = \frac{512}{15}
\]

Finally, multiply by \( 2\pi \):

\[
V = 2\pi \times \frac{512}{15} = \frac{1024\pi}{15}
\]

### Final Answer:
The volume of the solid is:

\[
V = \frac{1024\pi}{15} \, \text{cubic units}
\]

Would you like further details or clarifications?

---

Here are some related questions you might explore:

1. How does the disk method differ from the washer method?
2. What happens if the region is rotated about a different axis, like the y-axis?
3. Can the volume be computed using cylindrical shells instead?
4. What is the effect of changing the equation to \( y = 8 - 4x^2 \)?
5. How does the method work when revolving around a non-axis line?

**Tip**: Symmetry often simplifies integrals; use it when functions are even!

Find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 8 − 8x^2, y = 0

Explore the step-by-step process of finding the volume of the solid obtained by rotating the region bounded by y = 8 − 8x^2 and y = 0 around the x-axis. This problem uses the disk method and calculus concepts to find the volume, suitable for students in Grades 11-12.

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