Find the area of the parabola y² = 4ax bounded by it's latus rectum.

Concept:

Parabola:

• The focus of the parabola y² = 4ax is at (a, 0).
• The latus rectum of the parabola y² = 4ax cuts the parabola at (a, 2a) and (a, -2a).

Area under a curve:

• The area under the function y = f(x) from x = a to x = b and the x-axis is given by the definite integral \(\rm \left|\int_a^b f(x)\ dx\right|\), for curves which are entirely on the same side of the x-axis in the given range.
• If the curves are on both the sides of the x-axis, then we calculate the areas of both the sides separately and add them.

Calculation:

Since the graph of the parabola y² = 4ax is symmetrical about the x-axis, the required area is:

2 × \(\rm \left|\int_0^{a} \sqrt{4ax}\ dx\right|\) 

= 2 × 2√a \(\rm \int_0^{a} \sqrt x\ dx\) 

= 2 × 2√a \(\rm \left[\frac{2}{3}x^{\tfrac32}\right]_0^{a}\) 

= \(\rm {8\over3} \sqrt a\times a^{\tfrac32}\) = \(\rm 8a^2\over3\).

Additional Information:

The latus rectum is a line which passes through the focus and is parallel to the directrix.