Using integration, find the area of the region bounded between the line x = 2 and the parabola y^{2} = 8x.

Given equations are:

x = 2 ...... (1)

And y^{2} = 8x ...... (2)

Equation (1) represents a line parallel to y - axis at a distance of 2 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 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 (0,2) 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 = 2 and the parabola y^{2} = 8x is equal to square units.

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