Question

Find the area of the region bounded by the parabola y² = 4ax and the line x = a. 

Answer 1

\[\text{ The equation }y^2 = 4ax \text{ represents a parabola, with vertex at (0, 0) and symmetric about positive side of x - axis }\]
\[ x = \text{ a is a line parallel to } y -\text{ axis , and cutting x - axis at }(a, 0)\]
\[\text{ Making vertical strips of length } = \left| y \right|\text{ and width = dx in the quadrant OLSO . }\]
\[\text{ Area of approximating rectangle }= \left| y \right| dx\]
Since the approximating rectangle can move between x = 0 and x = a , 
\[\text{ and as the parabola is symmetric about } x -\text{ axis , }\]
\[\text{ Required shaded area OLSO }= A = 2 \times\text{ Area OLMO }\]
\[A = 2 \int_0^a \left| y \right| dx = 2 \int_0^a y dx ..............\left[ As, y > 0 \Rightarrow \left| y \right| = y \right]\]
\[ \Rightarrow A = 2 \int_0^a \sqrt{4ax} dx\]
\[ \Rightarrow A = 4 \int_0^a \sqrt{ax} dx\]
\[ \Rightarrow A = 4\sqrt{a} \int_0^a \sqrt{x} dx\]
\[ \Rightarrow A = 4\sqrt{a} \left[ \frac{x^\frac{3}{2}}{\frac{3}{2}} \right]_0^a \]
\[ \Rightarrow A = \frac{8}{3}\sqrt{a}\left[ a^\frac{3}{2} - 0 \right]\]
\[ \Rightarrow A = \frac{8}{3} a^2 \text{ sq . units }\]