Find the area bounded by the curves y^2=-4x and x^2=-4y

We are given two curves \( y^2 = -4x \) and \( x^2 = -4y \). Let's find the area bounded by these curves.

### Step 1: Rewrite the equations of the curves
First, let's express both equations in standard form:

- \( y^2 = -4x \): This represents a parabola that opens to the left.
- \( x^2 = -4y \): This represents a parabola that opens downward.

### Step 2: Symmetry and Intersection Points
These parabolas are symmetric with respect to the line \( y = x \). Let's find their points of intersection by solving the system of equations:

1. From \( y^2 = -4x \), solve for \( x \): \( x = -\frac{y^2}{4} \).
2. Substitute this into \( x^2 = -4y \).

Substituting:
\[
\left(-\frac{y^2}{4}\right)^2 = -4y
\]
Simplify:
\[
\frac{y^4}{16} = -4y
\]
Multiply through by 16:
\[
y^4 = -64y
\]
Factor out \( y \):
\[
y(y^3 + 64) = 0
\]
Thus, \( y = 0 \) or \( y^3 = -64 \), giving \( y = -4 \).

### Step 3: Find Corresponding \( x \)-values
For \( y = 0 \):
- From \( y^2 = -4x \), we get \( x = 0 \).
  
For \( y = -4 \):
- From \( y^2 = -4x \), we get \( (-4)^2 = -4x \), so \( 16 = -4x \), and \( x = -4 \).

### Step 4: Set up the Integral
We now know that the curves intersect at \( (0, 0) \) and \( (-4, -4) \).

The area between the curves can be calculated by setting up an integral in terms of \( y \). The right parabola \( y^2 = -4x \) gives \( x = -\frac{y^2}{4} \), and the left parabola \( x^2 = -4y \) gives \( x = -\sqrt{-4y} \).

The area between these curves from \( y = -4 \) to \( y = 0 \) is:
\[
A = \int_{-4}^{0} \left( -\frac{y^2}{4} - (-\sqrt{-4y}) \right) dy
\]

### Step 5: Calculate the Integral
Now we compute the integral:
\[
A = \int_{-4}^{0} \left( -\frac{y^2}{4} + 2\sqrt{-y} \right) dy
\]

Let me calculate this integral for you.from sympy import symbols, integrate, sqrt

# Define the variable and the expression to integrate
y = symbols('y')
expr = -y**2/4 + 2*sqrt(-y)

# Set the limits of integration
area = integrate(expr, (y, -4, 0))
area.simplify()The area bounded by the curves \( y^2 = -4x \) and \( x^2 = -4y \) is \( \frac{16}{3} \) square units.

Would you like more details on this calculation, or do you have any further questions?

Here are 5 related questions to explore:
1. How would the area change if the curves were shifted?
2. Can you find the area between similar curves opening in different directions?
3. What if one of the parabolas was rotated—how would that affect the area?
4. How do we calculate areas between curves using polar coordinates?
5. Can we apply this method to higher-dimensional problems, like volumes of revolution?

**Tip**: When calculating areas between curves, always ensure the limits of integration are correctly identified by finding their points of intersection.

Find the area bounded by the curves y^2 = -4x and x^2 = -4y.

Learn how to calculate the area bounded by the curves y^2 = -4x and x^2 = -4y using integration. This solution includes finding the points of intersection, setting up the integral, and calculating the area. Suitable for undergraduate-level calculus students.

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