Question

The area included between the parabolas \( y^2 = 5x \) and \( x^2 = 5y \) is

• A. \( \frac{25}{7} \, \text{sq. units} \)
• B. \( \frac{25}{3} \, \text{sq. units} \)
• C. \( \frac{25}{4} \, \text{sq. units} \)
• D. 25 sq. units

Hint:

To find the area between curves, solve for the points of intersection and set up an integral for the difference between the functions.

Answer 1

The Correct Option is B

Step 1: Set up the equations of the parabolas.
The equations of the parabolas are \( y^2 = 5x \) and \( x^2 = 5y \). To find the area between them, we first solve for \( y \) in terms of \( x \) for both equations.
Step 2: Find the points of intersection.
Next, we solve the system of equations \( y^2 = 5x \) and \( x^2 = 5y \) to find the points where the parabolas intersect. These points give the limits of integration for finding the enclosed area.

Step 3: Set up the integral.
We now compute the area using the formula for the area between curves. After performing the integration, we find that the area enclosed by the parabolas is \( \frac{25}{3} \) square units.
Step 4: Conclusion.
Thus, the correct answer is \( \frac{25}{3} \, \text{sq. units} \), corresponding to option (B).