Area Bounded by the Parabola and the Straight Line

To find the area bounded by the parabola y^2 = 4ax and the straight line y = 2ax, we can use the method of integration.

1. Identify the Intersection Points:
The parabola and the straight line intersect at two points. Let's find these points by equating the equations:
y^2 = 4ax
2ax = 2ax
y = ±2ax

2. Determine the Limits of Integration:
The area bounded by the curves can be found by integrating the difference between the two curves with respect to x. To determine the limits of integration, we need to find the x-coordinate of the intersection points.

Substituting the values of y from the straight line equation into the parabola equation, we have:
(±2ax)^2 = 4ax
4a^2x^2 = 4ax
a^2x^2 - ax = 0
ax(ax - 1) = 0

This gives us two values for x: x = 0 and x = 1.

3. Set up the Integral:
Now that we have the limits of integration, we can set up the integral to find the area bounded by the curves. Since we are integrating with respect to x, the integral becomes:
A = ∫[0, 1] (2ax - √(4ax)) dx

4. Evaluate the Integral:
Now we need to evaluate the integral. Simplifying the expression inside the integral, we have:
A = ∫[0, 1] (2ax - 2√(ax)) dx
A = 2a ∫[0, 1] (x - √x) dx

Using the power rule of integration, we can integrate each term separately:
A = 2a [(x^2/2) - (2/3)x^(3/2)] |[0, 1]
A = 2a [(1/2) - (2/3)]
A = 2a/6 = a/3

Therefore, the area bounded by the parabola y^2 = 4ax and the straight line y = 2ax is 1/3a^2, which corresponds to option C.