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

To find the area enclosed by the curves \( y^2 = 4 \cdot 21 \cdot x \) and \( x^2 = 4 \cdot 21 \cdot y \), we can proceed by analyzing their symmetry and intersections.

Firstly, rewrite the equations for clarity:
\[ y^2 = 84x \]
\[ x^2 = 84y \]

These equations can be interpreted as parabolas. The first one opens to the right, and the second one opens upwards.

1. **Find the points of intersection:**

To find the points of intersection, we solve the two equations simultaneously. Set \( y^2 = 84x \) into \( x^2 = 84y \):

\[ y^2 = 84x \implies x = \frac{y^2}{84} \]
\[ x^2 = 84y \implies \left(\frac{y^2}{84}\right)^2 = 84y \]
\[ \frac{y^4}{84^2} = 84y \]
\[ y^4 = 84^3 y \]
\[ y^4 - 84^3 y = 0 \]
\[ y(y^3 - 84^3) = 0 \]

So, \( y = 0 \) or \( y^3 = 84^3 \).

If \( y^3 = 84^3 \), then \( y = 84 \). Hence, \( y = 0 \) or \( y = 84 \).

For \( y = 0 \):
\[ y^2 = 84x \implies 0 = 84x \implies x = 0 \]

For \( y = 84 \):
\[ y^2 = 84x \implies 84^2 = 84x \implies x = 84 \]

Thus, the points of intersection are \( (0,0) \) and \( (84,84) \).

2. **Set up the integral:**

We will find the area by integrating the difference between the two functions from \( x = 0 \) to \( x = 84 \). 

From \( y^2 = 84x \), we get \( y = \sqrt{84x} \).

From \( x^2 = 84y \), we get \( y = \frac{x^2}{84} \).

The area \( A \) is given by:
\[ A = \int_{0}^{84} \left( \sqrt{84x} - \frac{x^2}{84} \right) dx \]

3. **Evaluate the integral:**

First, separate the integral:
\[ A = \int_{0}^{84} \sqrt{84x} \, dx - \int_{0}^{84} \frac{x^2}{84} \, dx \]

Calculate each part:
\[ \int_{0}^{84} \sqrt{84x} \, dx = \int_{0}^{84} 84^{1/2} x^{1/2} \, dx = 84^{1/2} \int_{0}^{84} x^{1/2} \, dx \]

\[ = 84^{1/2} \left[ \frac{2}{3} x^{3/2} \right]_{0}^{84} = 84^{1/2} \cdot \frac{2}{3} \cdot (84)^{3/2} \]

\[ = \frac{2}{3} \cdot 84^{1/2} \cdot 84^{3/2} = \frac{2}{3} \cdot 84^2 = \frac{2 \cdot 84^2}{3} = \frac{2 \cdot 7056}{3} = \frac{14112}{3} = 4704 \]

Now, the second integral:
\[ \int_{0}^{84} \frac{x^2}{84} \, dx = \frac{1}{84} \int_{0}^{84} x^2 \, dx \]

\[ = \frac{1}{84} \left[ \frac{x^3}{3} \right]_{0}^{84} = \frac{1}{84} \cdot \frac{84^3}{3} = \frac{84^2 \cdot 84}{3 \cdot 84} = \frac{84^2}{3} = \frac{7056}{3} = 2352 \]

Subtract the second result from the first:
\[ A = 4704 - 2352 = 2352 \]

Thus, the area enclosed by the curves is \( \boxed{2352} \).

Do you want more details or have any questions? Here are some related questions:

1. How do you find the area under a curve generally?
2. What are the steps to solve simultaneous equations?
3. What is the significance of finding points of intersection?
4. How do you determine the limits of integration for enclosed areas?
5. What is the relationship between integrals and areas under curves?
6. Can you explain why we separate the integral in this problem?
7. How do you handle integration involving square roots?
8. What other methods can be used to find areas between curves?

**Tip:** When solving integrals involving squares or square roots, substituting variables can simplify the problem significantly.

Learn how to find the area enclosed by the curves y^2 = 84x and x^2 = 84y with a detailed step-by-step solution. This problem covers calculus topics including integration of square root functions. Suitable for college-level students.

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