Sketch the region bounded by the curves y=4x3,y=4 and x=0
 then find the volume of the solid generated by revolving this region about the x
-axis.


We need to follow two steps:

1. **Sketch the region bounded by the curves:**
   The region is bounded by the following:
   - \( y = 4x^3 \)
   - \( y = 4 \)
   - \( x = 0 \) (the y-axis)

   To sketch this:
   - The curve \( y = 4x^3 \) is a cubic function that passes through the origin and increases as \( x \) increases.
   - The line \( y = 4 \) is a horizontal line.
   - The line \( x = 0 \) is the y-axis.

   The region is between these curves, from \( x = 0 \) to the point where \( y = 4x^3 \) intersects \( y = 4 \).

2. **Find the volume of the solid generated by revolving this region around the x-axis:**

   The method we will use here is the **disk method** because we are revolving around the x-axis. The formula for the volume using the disk method is:

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

   We need to determine the limits of integration \( a \) and \( b \), where the region is bounded along the x-axis. These can be found by solving the intersection of \( y = 4x^3 \) and \( y = 4 \):

   \[
   4x^3 = 4
   \]
   \[
   x^3 = 1 \implies x = 1
   \]

   So, the region is bounded between \( x = 0 \) and \( x = 1 \).

   The volume generated is given by:

   \[
   V = \pi \int_{0}^{1} [4]^2 - [4x^3]^2 \, dx
   \]

   Let’s calculate the volume.

\[
V = \pi \int_0^1 \left(16 - 16x^6 \right) dx
\]

Now we can evaluate this integral.

\[
V = \pi \left( \left[ 16x \right]_0^1 - \left[ \frac{16x^7}{7} \right]_0^1 \right)
\]

\[
V = \pi \left( 16 - \frac{16}{7} \right) = \pi \left( \frac{112}{7} - \frac{16}{7} \right)
\]

\[
V = \pi \times \frac{96}{7} = \frac{96\pi}{7}
\]

Thus, the volume of the solid is:

\[
V = \frac{96\pi}{7}
\]

### Would you like details on any step? Here are some related questions:

1. How does the disk method work for volume calculation?
2. What is the process for setting up the limits of integration?
3. Could we use the washer method in any way here?
4. Can you explain how to graph the region in more detail?
5. How does revolving around different axes change the volume?

**Tip:** Always ensure you set correct limits by finding where the curves intersect when dealing with volume of solids of revolution.

Sketch the region bounded by the curves y=4x^3, y=4, and x=0 then find the volume of the solid generated by revolving this region about the x-axis.

Learn how to calculate the volume of a solid generated by revolving the region bounded by the curves y=4x^3, y=4, and x=0 around the x-axis. This step-by-step guide uses the disk method, solving integrals and finding intersection points, ideal for students in Grades 11-12.

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