Find the area A bounded by the curves: x=y^4 and x=12y^3

To find the area \( A \) bounded by the curves \( x = y^4 \) and \( x = 12y^3 \), we first need to determine the points where the curves intersect, and then set up the appropriate integral.

### Step 1: Find the points of intersection
The points of intersection are found by equating the two curves:
\[
y^4 = 12y^3
\]
Dividing both sides by \( y^3 \) (assuming \( y \neq 0 \)):
\[
y = 12
\]
Also, \( y = 0 \) is a solution, as dividing by \( y^3 \) does not account for \( y = 0 \).

Thus, the curves intersect at \( y = 0 \) and \( y = 12 \).

### Step 2: Set up the integral for the area
The area between the curves can be found by integrating the difference between the rightmost curve (\( x = 12y^3 \)) and the leftmost curve (\( x = y^4 \)) over the interval from \( y = 0 \) to \( y = 12 \):
\[
A = \int_{0}^{12} (12y^3 - y^4) \, dy
\]

### Step 3: Compute the integral
Now, we evaluate the integral:
\[
A = \int_{0}^{12} (12y^3 - y^4) \, dy
\]
Breaking it up into two separate integrals:
\[
A = \int_{0}^{12} 12y^3 \, dy - \int_{0}^{12} y^4 \, dy
\]
Each term is integrated as follows:

- \( \int 12y^3 \, dy = 12 \cdot \frac{y^4}{4} = 3y^4 \)
- \( \int y^4 \, dy = \frac{y^5}{5} \)

Thus, the area becomes:
\[
A = \left[ 3y^4 - \frac{y^5}{5} \right]_{0}^{12}
\]

### Step 4: Evaluate the limits
Substitute \( y = 12 \) and \( y = 0 \) into the expression:

For \( y = 12 \):
\[
A = 3(12^4) - \frac{12^5}{5}
\]
Now compute \( 12^4 \) and \( 12^5 \):
- \( 12^4 = 20736 \)
- \( 12^5 = 248832 \)

Thus, the area is:
\[
A = 3(20736) - \frac{248832}{5} = 62208 - 49766.4 = 12441.6
\]

So, the area bounded by the curves is approximately \( \boxed{12441.6} \).

Would you like more details, or have any questions?

Here are 5 related questions:
1. How would the area change if the curves were rotated about the x-axis?
2. What is the significance of the points of intersection in relation to the area?
3. How do you determine which curve is the rightmost for setting up the integral?
4. Can you use polar coordinates to compute the area for these types of curves?
5. What happens if we calculate the volume by rotating the area around an axis?

**Tip:** Always sketch the curves before setting up integrals—it helps visualize which curve is on the top or right side.

Explore how to calculate the area bounded by the curves x = y^4 and x = 12y^3, using integral calculus. This step-by-step solution covers finding intersection points, setting up the integral, and evaluating the area. Suitable for high school and early college students.

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