Find the values of c such that the area of the region bounded by the parabolae y= x^2-c^2 and y=c^2-x^2 is 4608

To find the values of \( c \) such that the area of the region bounded by the parabolas \( y = x^2 - c^2 \) and \( y = c^2 - x^2 \) is 4608, we can follow these steps:

### Step 1: Find the points of intersection
The parabolas intersect where:
\[
x^2 - c^2 = c^2 - x^2
\]
Simplifying this:
\[
2x^2 = 2c^2 \quad \Rightarrow \quad x^2 = c^2 \quad \Rightarrow \quad x = \pm c
\]
So, the points of intersection are \( x = -c \) and \( x = c \).

### Step 2: Set up the area integral
The area between the curves from \( x = -c \) to \( x = c \) is given by:
\[
\text{Area} = \int_{-c}^{c} \left[ (c^2 - x^2) - (x^2 - c^2) \right] \, dx
\]
Simplifying the integrand:
\[
\text{Area} = \int_{-c}^{c} \left[ 2c^2 - 2x^2 \right] \, dx = 2 \int_{-c}^{c} \left[ c^2 - x^2 \right] \, dx
\]

### Step 3: Evaluate the integral
We can now evaluate the integral:
\[
\text{Area} = 2 \int_{-c}^{c} \left( c^2 - x^2 \right) dx
\]
This integral can be split as:
\[
\text{Area} = 2 \left[ \int_{-c}^{c} c^2 \, dx - \int_{-c}^{c} x^2 \, dx \right]
\]
\[
\text{Area} = 2 \left[ c^2 \int_{-c}^{c} 1 \, dx - \int_{-c}^{c} x^2 \, dx \right]
\]

Now, evaluate each integral:
\[
\int_{-c}^{c} 1 \, dx = 2c
\]
\[
\int_{-c}^{c} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-c}^{c} = \frac{c^3}{3} - \left(-\frac{c^3}{3}\right) = \frac{2c^3}{3}
\]

Substitute these back:
\[
\text{Area} = 2 \left[ c^2(2c) - \frac{2c^3}{3} \right] = 4c^3 - \frac{4c^3}{3} = \frac{12c^3 - 4c^3}{3} = \frac{8c^3}{3}
\]

### Step 4: Set the area equal to 4608
Now, set the area equal to 4608:
\[
\frac{8c^3}{3} = 4608
\]
Multiply both sides by 3:
\[
8c^3 = 13824
\]
Divide by 8:
\[
c^3 = 1728
\]
Take the cube root:
\[
c = \sqrt[3]{1728} = 12
\]

### Final Answer
The value of \( c \) such that the area of the region bounded by the parabolas is 4608 is \( \boxed{12} \).

Would you like a detailed explanation of any particular step, or do you have any questions?

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Here are 5 related questions to consider:

1. What is the area of the region bounded by the curves \( y = x^2 - 16 \) and \( y = 16 - x^2 \)?
2. How do you find the points of intersection between two curves?
3. What is the significance of symmetry in calculating areas bounded by curves?
4. How would the area change if the curves were \( y = x^2 - c^3 \) and \( y = c^3 - x^2 \)?
5. How can you verify the solution by considering the geometry of the parabolas?

**Tip:** When dealing with symmetric functions, you can often simplify the integration process by calculating the area for one half and then doubling the result.

Find the values of c such that the area of the region bounded by the parabolae y= x^2-c^2 and y=c^2-x^2 is 4608.

Explore how to determine the value of c for which the area bounded by the parabolas y = x^2 - c^2 and y = c^2 - x^2 equals 4608. This problem involves calculus concepts such as definite integrals, area between curves, and symmetry, making it suitable for students in Grades 11-12.

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