Find the area the region bounded by the parabola y^{2} = 4ax and the line x = a.

Given equations are:

x = a ...... (1)

And y^{2} = 4ax ...... (2)

Equation (1) represents a line parallel to the y - axis at a distance of units and equation (2) represents a parabola with vertex at origin and x - axis as its axis; A rough sketch is given as below: -

We have to find the area of the shaded region.

Required area

= shaded region OBAO

= 2 (shaded region OBCO) (as it is symmetrical about the x - axis)

(the area can be found by taking a small slice in each region of width Δx, then the area of that sliced part will be yΔx as it is a rectangle and then integrating it to get the area of the whole region)

(As x is between (0,a) and the value of y varies)

(as )

On integrating we get,

On applying the limits, we get,

Hence the area of the region bounded between the line x = a and the parabola y^{2} = 4ax is equal to square units.

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