Find the area under the curve y^2=4*21*x and x ^ 2 = 4*21*y

To find the area under the curve defined by the equations \( y^2 = 4 \cdot 21 \cdot x \) and \( x^2 = 4 \cdot 21 \cdot y \), we need to determine the region bounded by these curves and then calculate the integral over that region.

First, let's rewrite the given equations for clarity:
1. \( y^2 = 84x \)
2. \( x^2 = 84y \)

These equations represent a parabola and its reflection across the line \( y = x \).

### Step 1: Intersection Points

To find the area, we first need to determine the points where these curves intersect. Let's set \( y^2 = 84x \) equal to \( x^2 = 84y \):

\[ y^2 = 84x \]
\[ x^2 = 84y \]

Dividing one equation by the other:

\[ \frac{y^2}{x^2} = \frac{84x}{84y} \implies \left(\frac{y}{x}\right)^2 = \frac{x}{y} \]

Let \( t = \frac{y}{x} \). Then:

\[ t^2 = \frac{1}{t} \]
\[ t^3 = 1 \]

So, the solutions for \( t \) are:

\[ t = 1 \quad \text{(since we are looking for real intersections)} \]

This implies \( y = x \). Substituting \( y = x \) into either equation:

\[ x^2 = 84x \]

Thus,

\[ x(x - 84) = 0 \]

So, \( x = 0 \) or \( x = 84 \). Therefore, the points of intersection are \((0, 0)\) and \((84, 84)\).

### Step 2: Area Calculation

To find the area enclosed by these curves, we set up an integral. We will integrate with respect to \( x \) from \( 0 \) to \( 84 \), considering the upper function and the lower function:

The area \( A \) can be calculated as:

\[ A = 2 \int_{0}^{84} \sqrt{84x} \, dx \]

Here, \( \sqrt{84x} \) is derived from \( y^2 = 84x \), representing \( y \).

Perform the integral:

\[ A = 2 \int_{0}^{84} \sqrt{84x} \, dx \]
\[ A = 2 \int_{0}^{84} (84x)^{1/2} \, dx \]
\[ A = 2 \int_{0}^{84} 84^{1/2} x^{1/2} \, dx \]
\[ A = 2 \cdot 84^{1/2} \int_{0}^{84} x^{1/2} \, dx \]
\[ A = 2 \cdot \sqrt{84} \left[ \frac{2}{3} x^{3/2} \right]_{0}^{84} \]
\[ A = 2 \cdot \sqrt{84} \cdot \frac{2}{3} \left[ 84^{3/2} \right] \]
\[ A = \frac{4}{3} \sqrt{84} \cdot (84)^{3/2} \]
\[ A = \frac{4}{3} (84)^{1/2} \cdot (84)^{3/2} \]
\[ A = \frac{4}{3} \cdot 84^2 \]
\[ A = \frac{4}{3} \cdot 7056 \]
\[ A = \frac{28224}{3} \]
\[ A = 9408 \]

Thus, the area under the curve is \( \boxed{9408} \).

Do you want any details or have any questions? 

Here are some related questions:

1. How do you derive the formula for the area under a curve?
2. What is the significance of the intersection points in finding the area?
3. Can you explain how the method of integration helps in finding the area under curves?
4. What other methods can be used to find the area under a curve?
5. How does the symmetry of curves affect the area calculation?
6. Can you use numerical integration methods for this problem?
7. What are some real-world applications of finding the area under a curve?
8. How would the process change if the curves did not intersect?

**Tip:** When integrating functions involving radicals, it's helpful to simplify the expression before integrating to make the process easier.

Explore how to calculate the area under the curves defined by y^2 = 84x and x^2 = 84y using integral calculus. Detailed step-by-step solution provided, including intersection points and integration method. Suitable for advanced high school or college-level students.

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