Using integration, find the area of the region bounded between the line x = 2 and the parabola y2 = 8x.

Given equations are:

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


And y2 = 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: -


1.png


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 y2 = 8x is equal to square units.


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