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Find the area the region bounded by the parabola y2 = 4ax and the line x = a.
Given equations are:
x = a ...... (1)
And y2 = 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.
= 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)
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 y2 = 4ax is equal to square units.